Spin Representations of the Q-Poincaré Algebra

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Introduction 4

1 Construction of the q-Lorentz Algebra 8 1.1 q-Spinors and SUq(2)...... 9 1.1.1 q-SpinorsandTheirCotransformations ...... 9 1.1.2 The q-Spinor Metric and SLq(2)...... 10 1.1.3 Upper Spinor Indices, Conjugation, and SUq(2) ...... 10 1.2 The q-LorentzGroup...... 11 1.2.1 DottedSpinors ...... 11 1.2.2 Commutation Relations of the q-LorentzGroup ...... 12 1.3 The q-Lorentz Algebra as Dual of the q-LorentzGroup ...... 13 1.3.1 (su ) as dual of SU (2)...... 14 Uq 2 q 1.3.2 Computing the Dual of the q-LorentzGroup ...... 15

2 Structure of the q-Lorentz Algebra 18 2.1 Representation Theory of the q-LorentzAlgebra ...... 18 2.1.1 The Clebsch-Gordan Series of q(sl2)...... 18 2.1.2 Clebsch-Gordan Coeﬃcients ofU the q-Lorentz Algebra . . . 19 2.2 TensorOperators ...... 20 2.2.1 TensorOperatorsinHopfAlgebras ...... 20 2.2.2 Tensor Operators of q(su2) ...... 22 2.2.3 The Vector Form of U (su ) ...... 23 Uq 2 2.3 The q-LorentzAlgebraasQuantumDouble ...... 25 op 2.3.1 Rotations and the SUq(2) AlgebraofBoosts ...... 25 2.3.2 L-Matrices and the Explicit Form of the Boost Algebra . . 27 2.3.3 Commutation Relations between Boosts and Rotations . . 28 2.4 The Vectorial Form of the q-LorentzAlgebra ...... 30 2.4.1 Tensor Operators of the q-LorentzAlgebra ...... 30 2.4.2 TheVectorialGenerators...... 31 2.4.3 RelationswiththeotherGenerators...... 32

3 Algebraic Structure of the q-PoincaréAlgebra 34 3.1 The q-PoincaréAlgebra...... 34 3.1.1 Construction of the q-Minkowski-Space Algebra ...... 34 3.1.2 4-Vectors and the q-PauliMatrices ...... 35 3.1.3 Commutation Relations of the q-PoincaréAlgebra . . . . . 38 3.2 The q-Pauli-Lubanski Vector and the Spin Casimir ...... 40 3.2.1 The q-EuclideanAlgebra...... 40 3.2.2 The Center of the q-Euclidean Algebra ...... 42 3.2.3 The Pauli-Lubanski Vector in the q-Deformed Setting . . . 43 Contents 3

3.2.4 Boosting the q-Pauli-LubanskiVector ...... 45 3.3 TheLittleAlgebras...... 49 3.3.1 Little Algebras in the q-DeformedSetting ...... 49 3.3.2 Computation of the q-Little Algebras ...... 51

4 Massive Spin Representations 53 4.1 Representations in an Angular Momentum Basis ...... 53 4.1.1 The Complete Set of Commuting Observables ...... 53 4.1.2 Representations of the q-Euclidean Algebra ...... 55 4.1.3 Possible Transitions of Energy and Helicity ...... 56 4.1.4 Dependence on Total Angular Momentum ...... 57 4.1.5 Dependence on the other Quantum Numbers ...... 59 4.2 RepresentationsbyInduction ...... 64 4.2.1 The Method of Induced Representations of Algebras . . . . 64 4.2.2 Induced Representations of the q-Poincar´eAlgebra . . . . 65

5 Free Wave Equations 67 5.1 GeneralWaveEquations ...... 67 5.1.1 Wave Equations by Representation Theory ...... 67 5.1.2 q-LorentzSpinors ...... 68 5.1.3 ConjugateSpinors ...... 70 5.2 The q-DiracEquation...... 72 5.2.1 The q-DiracEquationintheRestFrame ...... 72 5.2.2 The q-Gamma Matrices and the q-Cliﬀord Algebra . . . . 73 5.2.3 The Zero Mass Limit and the q-Weyl Equations ...... 75 5.3 The q-MaxwellEquations ...... 76 5.3.1 The q-Maxwell Equations in the Momentum Eigenspaces . 76 5.3.2 Computing the q-MaxwellEquation...... 77 5.3.3 The q-Electromagnetic Field ...... 80

A Useful Formulas 82 A.1 Clebsch-GordanCoefficients ...... 82 A.1.1 Clebsch-Gordan and Racah Coefficients for q(su2) .... 82 A.1.2 Metric and Epsilon Tensor ...... U ...... 83 A.1.3 Clebsch-Gordan Coefficients for the q-LorentzAlgebra . . 86 A.2 Representations ...... 88 A.2.1 Representations of (su )...... 88 Uq 2 A.2.2 Representations of the q-LorentzAlgebra ...... 89 A.3 -matrices...... 90 A.3.1R The -Matrix of (su )...... 91 R Uq 2 A.3.2 The -Matrices of the q-LorentzAlgebra...... 92 R Bibliography 93 Introduction

Motivation From the beginnings of quantum field theory it has been argued that the pathological ultraviolet divergences should be remedied by limiting the precision of position measurements by a fundamental length [2–5]. In view of how position-momentum uncertainty enters into quantum mechanics, a natural way to integrate such a position uncertainty in quantum theory would have been to replace the commutative algebra of space observables with a non-commutative one [6]. However, deforming the space alone will in general break the symmetry of spacetime. In order to preserve a background symmetry the symmetry group must be deformed together with the space it acts on. It is clear that Lie groups cannot be continuously deformed within their proper category: From the classification of semi-simple Lie groups we know that they form a countable and hence discrete set. Being manifolds, however, they can be naturally embedded in the category of algebras by the Gelfand-Neumark map [7], the additional group structure on the manifold side translating into a Hopf structure on the algebra side. But although Hopf algebras in general had been familiar to mathematicians for some time [8–10], hardly any non-trivial examples of Hopf algebras were known [11]. This situation changed with the discovery of quantum groups [12], that is, with the discovery of generic methods to continuously deform Lie algebras [13,14] and matrix groups [15–17] within the category of Hopf algebras. Quantum groups now provided a consistent mathematical framework to for- mulate physical theories on non-commutative spaces. Beginning with the non- commutative plane [18], q-deformations of a variety of objects have since been constructed: differential calculi on non-commutative spaces [19], Euclidean space [16], Minkowski-Space [20], the Lorentz group and the Lorentz algebra [21–24], the Poincaréalgebra [25], to name a few. The study of these objects has produced interesting results. For example, it has been found that free theories on non- commutative spaces can be viewed as theories on ordinary commutative spaces with complicated interactions [26–28]. Another result is, that q-deformation will in general discretize the spectra of spacetime observables [29–31], that is, q-deformation puts physics on a spacetime lattice. This nourishes the hope that q-deformed field theories might be regular- ized, one of the original motivations to consider non-commutative geometries. It is not new that deformation is a method to regularize field theory — at least, it is one way to look at the first step of renormalization: In the loop expansion of transition probabilities some terms turn out to be infinite, so we regularize Introduction 5 them by a sort of deformation process in order to classify the divergences. In this sense q-deformation can be viewed as attempt to shift the deformation from the end of the construction of field theory (perturbative expansion) to the beginning (symmetry structures).

Aims Given the q-Poincaréalgebra as background symmetry, how can we construct a quantum theory upon it? If states continue to be described by vectors of a Hilbert space, it must be specified how the q-Poincaréalgebra acts on them, that is, we must construct representations of the q-Poincare algebra. If we further want to describe elementary particles the representations must be irreducible [1]. If we want to use reducible representations such as the Dirac spinor representation, we need additional constraints to eliminate the redundant degrees of freedom. These constraints are the wave equations. The interpretation of this quantum theory forces us to consider multi-particle states. These are described as properly symmetrized (or anti-symmetrized) tensor product representations. Symmetrization or anti-symmetrization means that we need a ray representation of the permutation group on the tensor product space which is compatible with (intertwines with) the action of the q-Poincaréalgebra. The physical states are the orbits of this action of the permutation group, while the direct sum of all such multi-particle spaces is the Fock space. We summarize:

(i) Elementary q-particles are irreducible representations of the q-Poincar´ealgebra.

(ii) Wave equations are the constraints to eliminate the redundant degrees of freedom of a reducible representation.

(iii) q-Fields are symmetrized or anti-symmetrized multi-particle states.

In the undeformed case these principles completely determine the free relativistic quantum theory. Therefore, it is reasonable to use them as program to construct the deformed theory. This program has been pursued in previous work [31–40]. In [31–35] irreducible spin zero representations of the q-Poincaréalgebra were constructed. While in [31, 32] the realization of the q-Poincaréalgebra within q-Minkowski phase space was considered, such that the representations were naturally lim- ited to orbital angular momentum,2 it is possible to extend [34, 35] to include spin representations (Sec. 4.1.1). Various methods to construct wave equations have been proposed, based on q-Clifford algebras [36], q-deformed co-spinors [37], or differential calculi on quantum spaces [38–40], leading to mutually different results. This is unsatisfactory since the construction of wave equations according to (ii) should determine the wave equations uniquely as in the undeformed

2See Eq. (3.57). 6 Introduction case [41] and should not demand any additional mathematical structure besides the q-Poincar´ealgebra and the basic apparatus of quantum mechanics. The aim of the present work is to investigate the nature of spin within the representation theory of the q-Poincar´ealgebra.

Results Our main results are:

The q-deformed Pauli-Lubanski vector is computed (Sec. 3.2), from which • the spin Casimir and the little algebras can be determined (Sec. 3.3).

Irreducible representations with spin are constructed (Chap. 4). • A practical method to uniquely compute the wave equations is developed • (Sec. 5.1). As examples the q-Dirac equation (Sec. 5.2) and the q-Maxwell equations (Sec. 5.3) are computed.

To give a more detailed overview: In chapter 1 we review the construction of the q-Lorentz algebra. We start with the quantum plane, xy = qyx, derive the algebra of coacting quantum matrices Mq(2), introduce the q-spinor metric, the quantum special linear group SLq(2) and its real form SUq(2). We introduce dotted spinors, join an undotted and a dotted corepresentation to form the quantum Lorentz group SLq(2, C). Using the duality between SLq(2) and q(sl2) we compute the quantum Lorentz algebra (sl (C)) by dualizing SL (2,UC) [18,20,22,42]. The presentation puts Uq 2 q emphasis on the fact that in the construction of the q-Lorentz algebra as it is understood now, hardly any arbitrariness is involved. Chapter 2 explores the structure of the q-Lorentz algebra. The representation theory of the q-Lorentz algebra is reviewed, explicit formulas for the q-Clebsch- Gordan coefficients are given. After a general consideration of the different sorts of tensor operators, the vectorial generators of q(sl2) are determined. Three different forms of the q-Lorentz algebra are relatedU by explicit formulas: the dual of the q-Lorentz group [42], the quantum double form [21], and the vectorial or RS-form [23, 43, 44]. The isomorphism between the quantum double form and the vectorial form that is found (Sec. 2.4.3) relates the work of the Warsaw and the Munich group. In chapter 3 the results of chapter 2 are used to construct the algebra of q- Minkowski space [20]. Commutation relations of the generators of different forms of the q-Lorentz algebra with the spacetime generators are given. We study the structure of the q-Euclidean algebra consisting of rotations and translations in order to find a good zero component of the q-Pauli-Lubanski vector. A technique of boosting is used to calculate the other components (Sec. 3.2). The q-Pauli- Lubanski vector is used to compute the little algebras for the massive and the massless case (Sec. 3.3). Introduction 7

Chapter 4 contains the construction of massive spin representations of the q-Poincaréalgebra. In the first part we construct irreducible representations in an angular momentum basis, which is accessible to physical interpretation. The calculations are considerably simplified by the q-Wigner-Eckart theorem. In the second part we briefly show how representations of the q-Poincaréalgebra can be constructed using the method of induced representations. In chapter 5 we calculate free wave equations. We start with the representation theoretic interpretation of free wave equations. Then we consider the generalities of q-Lorentz spinor representations, conjugate spinors, and the relation between momenta and derivations. Finally, we put things together and uniquely determine the q-Dirac equation including q-gamma matrices and their q-Clifford algebra, the q-Weyl equations, and the q-Maxwell equations.

Outlook While our approach to the q-Poincar´ealgebra was representation theoretic, the problems we had to overcome were mostly on the algebraic side: A method to boost vector operators, complete sets of commuting observables, the spin Casimir, the spin symmetry algebras, spinor conjugation — all this had to be found before spin representations and spinorial wave equations could be computed. Now, that the algebraic tool set is more complete, we are prepared for the next steps towards a q-deformed relativistic quantum theory. One promising way to continue this work would be to couple the q-Dirac and the q-Maxwell ﬁeld, for which the mathematical setting has been provided in chapter 5.

Notation Throughout this work, the deformation parameter q is assumed real, q > 1. We frequently use the abbreviations

qj q−j λ := q q−1 , [j] := − , (1) − q q−1 − where j is a number. In particular, we have [2] = q + q−1. Spinor indices running through , + = 1 , + 1 are denoted by lower case Roman letters (a, b, c, d), {− } {− 2 2 } 3-vector indices running through , 3, + = 1, 0, +1 by upper case Roman letters (A, B, C), and 4-vector indices{− running} {− through }0, , +, 3 by lower case Greek letters (µ, ν, σ, τ). Quantum Lie groups are written{ − with} a subscript q like SL (2), quantum enveloping algebras like (sl ). q Uq 2 Chapter 1 Construction of the q-Lorentz Algebra

In undeformed quantum mechanics we can represent a state by a wave function n ψ : R C. In this representation, the observables xi, which describe the measurement→ of the position of the particle, act on ψ by multiplications with the functions x : Rn C, x (~r) = r . In this sense, geometry is described i → i i by the algebra of functions over a space, (Rn), rather than by the space Rn itself. Replacing a space by its functionF algebra, it is natural to replace an endomorphism f by its pullback f ∗,

f f ∗ Rn Rn (Rn) (Rn) , where (f ∗x )(~r) := x (f~r) , (1.1) −→ ⇒ F ←− F i i yielding a recipe to translate spaces and homomorphisms of spaces into algebras and homomorphisms of algebras. In the language of category theory is called a cofunctor [45], the preﬁx “co” reminding us that we have to reverseF arrows. For a consistent mathematical framework we must extend this method of algebraization to any additional structure on Rn. If there is for example the action φ of a group G on the space we get

φ φ∗ G Rn Rn (G) (Rn) (Rn) , (1.2) ⊗ −→ ⇒ F ⊗F ←− F where (G) is the algebra of functions over the group and the homomorphism F of algebras ρ := φ∗ is called the coaction. The structure maps of the group, multiplication µ, unit η, and inverse, translate into comultiplication ∆ = µ∗, counit ε = η∗, and coinverse or antipode S. The group axioms translate into axioms of this co-structure [8]. An algebra equipped with this co-structure is called a Hopf algebra [9, 10]. So far, the structure of spaces and groups acting on them has only been rephrased in a more algebraic but equivalent language. But unlike the category of Lie groups, the category of algebras allows for continuous deformation: We can replace the trivial commutation relations of the algebra of space functions by non-trivial ones, which depend on a real parameter q. This q-deformation of the space algebra forces us to q-deform any Hopf algebra coacting on it, as well. Reminiscent of their relation to quantum theory, these deformed algebras are called quantum spaces and quantum groups. Instead of quantum groups we can consider their Hopf duals [46, 47], the quantum algebras, which are deformations of the enveloping Lie algebras. Since quantum algebras have a familiar 1.1 q-Spinors and SUq(2) 9 undeformed counterpart, they become directly accessible to physical interpretation. For example, the generators of the quantum algebra of rotations are the q-deformed angular momentum operators.

1.1 q-Spinors and SUq(2) 1.1.1 q-Spinors and Their Cotransformations The simplest quantum space is the deformation of the algebra (C2) = C[x, y] of polynomial spinor functions. We replace the trivial commutationF relations xy = yx with xy = qyx, where q is a real parameter q > 1, and call the resulting algebra C2 := C x, y / xy = qyx (1.3) q h i h i the algebra of q-spinors or the quantum plane [18]. As in the undeformed case we want the spinor algebra to carry a left and a right matrix corepresentation. We deﬁne the vector of spinor generators

ψa =(ψ−, ψ+) := (x, y) (1.4) a matrix of generators of the algebra M (2) of 2 2-matrices q × a b M a = (1.5) b c d with respect to the indices , + = 1 , + 1 and the left and right coaction of {− } {− 2 2 } this matrix on the spinor

a a′ ρ (ψ ) := M ′ ψ ′ , ρ (ψ ) := ψ ′ M , (1.6) L a a ⊗ a R a a ⊗ a where we sum over repeated indices, and where the coproduct of Mq(2) is deﬁned by ∆(M a )= M a M b . c b ⊗ c We want the deformed commutation relations between the generators of Mq(2) to be consistent with those of the q-spinor, xy = qyx, that is, the coaction maps must be algebra homomorphisms. This uniquely determines the relations ab = qba , ac = qca, bd = qdb, cd = qdc (1.7) bc = cb, ad da =(q q−1)bc . − − The algebra freely generated by a,b,c,d modulo these relations (1.7) is called M (2) the algebra of 2 2 quantum matrices. Introducing the R-Matrix q × q 0 00 ab 0 1 00 R cd =   (1.8) 0 q q−1 1 0 − 0 0 0 q     10 1. Construction of the q-Lorentz Algebra with respect to the indices , +, + , ++ , Eqs. (1.7) can be written in the compact form {−− − − }

ab c′ d′ b a a′b′ R c′d′ M cM d = M b′ M a′ R cd , (1.9) the famous FRT-relations of matrix quantum groups [16].

1.1.2 The q-Spinor Metric and SLq(2) With the spinor metric 0 q−1/2 ε = εab = , with ε εbc = δc (1.10) ab − q1/2 0 ab a − ab we can write xy = qyx as ψaψbε = 0. In analogy to the undeformed case the spinor metric must thus be invariant under Mq(2)-transformations up to a factor. Indeed, we ﬁnd

a b a′b′ ab M a′ M b′ ε = (detqM) ε , (1.11) where detqM = ad qbc is central in Mq(2). Constraining the transformations − ab to leave the scalar product ψaφbε of two spinors strictly invariant, we obtain SLq(2) := Mq(2)/ detqM = 1 the deformation of the function algebra of the group of special linearh transformations.i Finally, Eq. (1.11) can be contracted with the metric from the right. From the resulting equation

′ ′ a ′ b ′ a b a M a (M b ε εbc)= δc (1.12) we can read oﬀ the antipode

−1 a aa′ b′ d q b S(M b) := ε M a′ εb′b = − , (1.13) qc a − −1 a a playing the role of the inverse, (M ) b = S(M b). This completes the Hopf algebra structure of SLq(2).

1.1.3 Upper Spinor Indices, Conjugation, and SUq(2)

Deﬁning a transposition on SLq(2) by

a T a b T (M b)=(M ) b := M a , (1.14) we can consider now a spinor transforming under the congredient representation (M T )−1. As in the undeformed case we indicate this transformation property by an upper index.

a b T −1 b b a b aa′ b′ ρ (ψ )= ψ ((M ) ) = ψ S(M )= ψ ε M ′ ε ′ (1.15) R ⊗ a ⊗ b ⊗ a b b 1.2 The q-Lorentz Group 11

Contracting this equation with the spinor metric we ﬁnd

a′ b′ b ρ (ε ′ ψ )=(ε ′ ψ ) M , (1.16) R aa bb ⊗ a a′ telling us that εaa′ ψ transforms as a spinor with lower index. We conclude that we can raise and lower indices by

a aa′ a′ ψ = ε ψa′ , ψa = εaa′ ψ . (1.17)

When we rewrite the spinor commutation relations as

ab a′ b′ ab a b a b ba 0= ψ ψ ε = ε ′ ε ′ ψ ψ ε = ψ ψ ε = ψ ψ ε , (1.18) a b aa bb ba − we see that a spinor with upper index satisﬁes commutation relations opposite to a spinor with lower index. Thus, we can deﬁne a -structure on the spinor algebra C2 by (ψ )∗ := ψa. This induces a -structure∗ on SL (2) as well, by q a ∗ q demanding the stars to be compliant with the coaction, ρR =( ) ρR. A stared spinor transforms as a spinor with upper index,◦∗ that∗⊗∗ is,◦ by the congredient representation. We conclude that the induced -operation on SLq(2) is given by ∗

a ∗ b (M b) = S(M a) . (1.19)

In other words, we have (M T )∗ = M −1, which can be viewed as a quantum group analogue of a unitarity condition. Therefore, SL (2) with this -structure q ∗ is called SUq(2).

1.2 The q-Lorentz Group 1.2.1 Dotted Spinors We want to construct a deformation of the Lorentz group SL(2, C), which is, viewed as real manifold, 6-dimensional, having 6 independent inﬁnitesimal generators. Now, a spinor and its complex conjugate and thus the corepresentation matrix and its conjugate are no longer linearly dependent. This means that we a a ∗ ∗ have to add the conjugates M¯ b := (M b) and ψ¯a := (ψa) as extra generators. Of course, the conjugate spinor cotransforms under the conjugate matrix. As in the undeformed case, we will indicate that a quantity transforms like a conjugate spinor by a dotted index. Thus, writing ψa˙ implies

b ρ (ψ )= ψ˙ M¯ , (1.20) R a˙ b ⊗ a where we think of the dot as belonging to ψ rather than to the index itself. Since the -operation is by definition an algebra anti-homomorphism (and a coalgebra homomorphism),∗ the conjugate generators satisfy the opposite commutation 12 1. Construction of the q-Lorentz Algebra relations of their pre-images. However, it is more convenient to combine the conjugate generators M¯ linearly to form another matrix M2 defined implicitly by T −1 ¯ (M2 ) := M, that is, b ¯ a S(M2 a) := M b . (1.21) S T is an algebra anti-homomorphism (and a coalgebra homomorphism), so M ◦ 2 naturally generates a SLq(2) Hopf algebra. We now have two sets of generators generating two copies of SLq(2). For a consistent notation we will subscript the first set M = M as well. The -operation can then be written as 1 ∗ a ∗SLq(2,C) a ∗SUq(2) (M1 b) =(M2 b) . (1.22) Finally, we introduce upper dotted indices by demanding them to transform according to ˙ ρ (ψa˙ )= ψb M b . (1.23) R ⊗ 2 a This leads to formulas for raising and lowering dotted indices

a˙ ba b˙ ψ = ψb˙ ε , ψa˙ = ψ εba . (1.24)

1.2.2 Commutation Relations of the q-Lorentz Group So far, we know that the q-Lorentz group must be generated by two copies of a a SLq(2), generated by two sets of generators M1 b and M2 b, respectively. The only thing we do not know yet are the commutation relations between M1 and M2. A priori, there are several choices of commutation relations, from which we will select one by an additional requirement: We will demand SLq(2, C) to possess a substructure of rotational symmetry, that is, we are looking for a homomorphism of Hopf- algebras1 µ : SL (2, C) SU (2). ∗ q → q Embedding the generators by M a ֒ M a 1 and M a ֒ 1 M a in a 1 b → b ⊗ 2 b → ⊗ b tensor product of two SLq(2), the multiplication map µ : SL (2) SL (2) SU (2) (1.25) q ⊗ q → q is the obvious choice. Note, that according to the preceding section g h SL (2) SL (2) is to be equipped with the -structure (g h)∗ := h∗ g∗⊗, such∈ q ⊗ q ∗ ⊗ ⊗ that µ((g h)∗)= µ(h∗ g∗) = h∗g∗ = (gh)∗ = µ(g h)∗. In other words, µ is already compliant⊗ with the⊗ -structures. ⊗ ∗ For µ to be a homomorphism of algebras, the images of the generators, a a a µ(M1 b) = µ(M2 b) = M b, must satisfy the SLq(2) commutation relations (1.9). This means that the generators have to satisfy

′ ′ ′ ′ ab ′ ′ c d b ′ a ′ a b R c d M2 cM1 d = M1 b M2 a R cd , (1.26)

1During the transition from groups to quantum groups the arrows of mappings have to be reversed. 1.3 The q-Lorentz Algebra as Dual of the q-Lorentz Group 13 which completes the algebraic structure of the q-Lorentz group.2 To summarize, let us give a compact and rigorous deﬁnition of the q-Lorentz group [21, 22]. First we need to deﬁne the so-called coquasitriangular map R : SL (2) SL (2) C on the generators by q ⊗ q → a b − 1 ab R(M c, M d) := q 2 R cd , (1.27) which can be shown to extend to all of SLq(2) by linearity in both arguments and by demanding

R(fg,h) := R(f, h(1))R(g, h(2)) , R(f,gh) := R(f(1), h)R(f(2),g) . (1.28)

− 1 The factor q 2 has been introduced for convenience. The map R has a unique convolution inverse, that is, a map R−1 : SL (2) SL (2) C with q ⊗ q → −1 −1 R(a(1), b(1))R (a(2), b(2))= R (a(1), b(1))R(a(2), b(2))= ε(a)ε(b) , (1.29) simply deﬁned by

−1 a b 1 −1 ab R (M c, M d) := q 2 (R ) cd . (1.30)

Using R, the commutation relations of SLq(2) can be written as

R(a(1), b(1))a(2)b(2) = b(1)a(1)R(a(2), b(2)) . (1.31)

−1 Deﬁnition 1. Let R denote the coquasitriangular map of SLq(2) and R its convolution inverse. The vector space SLq(2) SLq(2) with tensor product coalgebra structure, ∆(g h)=(g h ) (g ⊗ h ), ε(g h)= ε(g)ε(h), with ⊗ (1) ⊗ (1) ⊗ (2) ⊗ (2) ⊗ multiplication (g h)(g′ h′)= gg′ h h′ R−1(h ,g′ )R(h ,g′ ) (1.32) ⊗ ⊗ (2) ⊗ (2) (1) (1) (3) (3) antipode S(g h)=(1 S(h))(S(g) 1), and -structure ⊗ ⊗ ⊗ ∗ (g h)∗SLq(2,C) = h∗SUq(2) g∗SUq(2) (1.33) ⊗ ⊗ is the q-Lorentz group SLq(2, C).

1.3 The q-Lorentz Algebra as Dual of the q-Lorentz Group For a symmetry of a quantum mechanical system the mathematical object with a direct physical interpretation is the enveloping algebra of the symmetry group’s Lie algebra rather than the group itself. The Hilbert space representations of its generators are the observables of the conserved quantities corresponding to the symmetry. Consequently, rather than in the quantum group itself we are interested in its dual, the quantum enveloping algebra. 2If we drop the requirement of a subsymmetry of rotations, we can construct an alternative q-Lorentz group with two commuting copies of SLq(2). It turns out to be unphysical, however, insofar as the according q-Poincar´ealgebra has no mass Casimir. 14 1. Construction of the q-Lorentz Algebra

1.3.1 Uq(su2) as dual of SUq(2) We will call two Hopf- algebras A and H dual to each other if there is a dual pairing [46] between them:∗

Deﬁnition 2. Let A and H be Hopf- algebras. A non-degenerate bilinear map ∗ , : A H C , (a, h) a, h (1.34) h· ·i × −→ 7−→ h i is called a dual pairing of A and H if it satisﬁes

(i): ∆(a),g h = a, gh , a b, ∆(h) = ab, h h ⊗ i h i h ⊗ i h i (ii): a, 1 = ε(a) , 1, h = ε(h) (1.35) h i h i (iii): a∗, h = a, (Sh)∗ . h i h i Remark that for property (i) we have to extend the dual pairing on tensor products by

a b, g h := a, g b, h . (1.36) h ⊗ ⊗ i h ih i From the properties of the dual pairing it follows that

S(a), h = a, S(h) . (1.37) h i h i

The following algebra is dual to SUq(2)

Deﬁnition 3. The algebra generated by E, F , K, and K−1 with commutation relations KK−1 = K−1K =1 and

K K−1 KE = q2EK, KF = q−2FK, EF F E = − , (1.38) − q q−1 − Hopf structure

∆(E)= E K +1 E, S(E)= EK−1 , ε(E)=0 ⊗ ⊗ − ∆(F )= F 1+ K−1 F, S(F )= KF, ε(F ) = 0 (1.39) ⊗ ⊗ − ∆(K)= K K, S(K)= K−1 , ε(K)=1 ⊗ and -structure ∗ E∗ = FK, F ∗ = K−1E, K∗ = K (1.40) is called (su ), the q-deformation of the enveloping algebra (su ) [48,49]. Uq 2 U 2 1.3 The q-Lorentz Algebra as Dual of the q-Lorentz Group 15

The dual pairing of (su ) and SU (2) is deﬁned on the generators by Uq 2 q

− 1 −1 a 0 0 a 0 q 2 a q 0 E, M b := 1 , F, M b := , K, M b := . h i q 2 0 h i 0 0 h i 0 q (1.41)

There is a universal -matrix (Sec. A.3) for q(sl2) deﬁned by the formal power series R U

∞ = q(H⊗H)/2 R (q)(En F n) (1.42) R n ⊗ n=0 X where R (q) := qn(n−1)/2(q q−1)n([n]!)−1, and K = qH [12]. It is dual to the n − coquasitriangular map R of SUq(2) in the sense that

,g h = R(g, h) (1.43) hR ⊗ i for all g, h SUq(2). This duality is the reason why we have introduced the factor − 1 ∈ q 2 in the deﬁnition (1.27) of the coquasitriangular map R. We will sometimes write in a Sweedler like notation = [1] [2], where the subscripts stand for an index which is summed over. R R ⊗ R

1.3.2 Computing the Dual of the q-Lorentz Group The map of the dual pairing of (sl ) and SL (2) naturally extends to a pairing Uq 2 q of the tensor product spaces q(sl2) q(sl2) and SLq(2, C) ∼= SLq(2) SLq(2) by U ⊗ U ⊗

a b, g h := a, g b, h (1.44) h ⊗ ⊗ i h ih i for all a, b (sl ) and g, h SL (2). By construction, this pairing is non- ∈ Uq 2 ∈ q degenerate. We now want to deﬁne a Hopf algebra structure on q(sl2) q(sl2) which turns this into a pairing of Hopf algebras. Firstly, the multiplicatU ⊗ion U must satisfy

(a b)(a′ b′),g h =! (a b) (a′ b′), ∆(g h) h ⊗ ⊗ ⊗ i h ⊗ ⊗ ⊗ ⊗ i = a a′, ∆(g) b b′, ∆(h) = aa′,g bb′, h h ⊗ ih ⊗ i h ih i = aa′ bb′,g h . (1.45) h ⊗ ⊗ i Hence, the multiplication on the vector space (sl ) (sl ) must be defined Uq 2 ⊗ Uq 2 by (a b)(a′ b′) = aa′ bb′, which means that as an algebra the dual of the q-Lorentz⊗ group⊗ is just the⊗ tensor algebra of two copies of (sl ). Secondly, we Uq 2 16 1. Construction of the q-Lorentz Algebra want to define a coproduct that is consistent with the pairing. ∆(a b), (g h) (g′ h′) =! a b, (g h)(g′ h′) h ⊗ ⊗ ⊗ ⊗ i h ⊗ ⊗ ⊗ i = a b,gg′ h h′ R−1(h ,g′ )R(h ,g′ ) h ⊗ (2) ⊗ (2) i (1) (1) (3) (3) = ∆(a),g g′ ∆(b), h h′ −1, h g′ , h g′ h ⊗ (2)ih (2) ⊗ ihR (1) ⊗ (1)ihR (3) ⊗ (3)i = a ,g −1, h b , h , h h (1) ihR[1] (1)ih (1) (2)ihR[1] (3)i −1,g′ a ,g′ ,g′ b , h′ × hR[2] (1)ih (2) (2)ihR[2] (3)ih (2) i = a ,g −1b , h −1a ,g′ b , h′ h (1) ihR[1] (1)R[1] ihR[2] (2)R[2] ih (2) i = (a −1b ) ( −1a b ), (g h) (g′ h′) (1.46) h (1) ⊗ R[1] (1)R[1] ⊗ R[2] (2)R[2] ⊗ (2) ⊗ ⊗ ⊗ i From the last line we read off the coproduct ∆(a b)= −1∆⊗2(a b) , (1.47) ⊗ R23 ⊗ R23 where =1 1 and ∆⊗2(a b)=(a b ) (a b ). This tells us, that R23 ⊗R⊗ ⊗ (1)⊗ (1) ⊗ (2) ⊗ (2) the coproduct of the q-Lorentz algebra is the tensor coproduct of q(sl2) q(sl2) with the two inner tensor factors twisted by the universal -matrix.U ⊗U R Thirdly, the same reasoning for the antipode S(a b),g h =! a b, S(g h) = a b, (1 Sh)(Sg 1) h ⊗ ⊗ i h ⊗ ⊗ i h ⊗ ⊗ ⊗ i = a b, (Sg) (Sh) R−1 (Sh) , (Sg) R (Sh) , (Sg) h ⊗ (2) ⊗ (2)i (1) (1) (3) (3) = a, S(g ) b, S(h ) R−1 h ,g R h ,g h (2) ih (2) i (3) (3) (1) (1) = ,g S(a),g −1,g , h S(b), h −1, h hR[2] (1)ih (2)ihR[2] (3)ihR[1] (1)ih (2)ihR[1] (3)i = S(a) −1,g S(b) −1, h hR[2] R[2] ihR[1] R[1] i = S(a) −1 S(b) −1,g h (1.48) hR[2] R[2] ⊗ R[1] R[1] ⊗ i leads to S(a b)= (Sa Sb) −1 , (1.49) ⊗ R21 ⊗ R21 where = . The antipode is the tensor antipode twisted by the R21 R[2] ⊗ R[1] transposed universal R-matrix. The counit ε(a b) = ε(a)ε(b) follows directly from the definition of the pairing. Finally, we⊗ need to calculate the star structure. (a b)∗,g h =! a b, S((g h)∗) = S(a b), h∗ g∗ h ⊗ ⊗ i h ⊗ ⊗ i h ⊗ ⊗ i = S(a) −1, h∗ S(b) −1,g∗ hR[2] R[2] ihR[1] R[1] i = a∗ −1, h b∗ −1,g (1.50) hR[1] R[1] ihR[2] R[2] i Here we have used that is real, ∗⊗∗ = . Thus, we find R R R21 (a b)∗ = (b∗ a∗) −1 , (1.51) ⊗ R21 ⊗ R21 which completes the structure of the q-Lorentz algebra. To summarize, we have the following 1.3 The q-Lorentz Algebra as Dual of the q-Lorentz Group 17

Proposition 1. The tensor product algebra q(sl2) q(sl2) with the Hopf- - structure U ⊗ U ∗ ∆(a b)= −1∆⊗2(a b) , S(a b)= (Sa Sb) −1 ⊗ R23 ⊗ R23 ⊗ R21 ⊗ R21 (1.52) ε(a b)= ε(a)ε(b) , (a b)∗ = (b∗ a∗) −1 ⊗ ⊗ R21 ⊗ R21 is the Hopf- -dual of the q-Lorentz group SLq(2, C). Therefore, we will call it the q-Lorentz algebra∗ (sl (C)) [42]. Uq 2 There are two universal -matrices of the q-Lorentz algebra, which are com- R posed of the -matrix of (sl ) according to R Uq 2 = −1 −1 , = −1 . (1.53) RI R41 R31 R24R23 RII R41 R13R24R23 is anti-real while is real. RI RII Chapter 2 Structure of the q-Lorentz Algebra

2.1 Representation Theory of the q-Lorentz Algebra

2.1.1 The Clebsch-Gordan Series of Uq(sl2)

Let us review some facts about the representation theory of q(sl2) and q(su2) [50]. For any j 1 N there is an irreducible representationU on the (2Uj + 1)- ∈ 2 0 dimensional Hilbert space Dj with orthonormal basis j, m m = j, j + 1,...j and representation map ρj : (sl ) Aut(Dj) given{| byi| 1 − − } Uq 2 → ρj(E) j, m = q(m+1) [j + m + 1][j m] j, m +1 | i − | i ρj(F ) j, m = q−m p[j + m][j m + 1] j, m 1 (2.1) | i − | − i ρj(K) j, m = q2m j, m . | i |p i 0 For the real form q(su2) these are even -representations. D is called the scalar U1 ∗ 1 representation, D 2 the fundamental or spinor representation, and D the vector representation. Recall that the coproduct of a Hopf algebra enables us to construct the tensor j j′ product of two representations: Let D and D be representations of q(sl2) as ′ deﬁned above, with representation maps ρj and ρj . Then there is a representationU ′ ′ on the tensor product space Dj Dj with representation map (ρj ρj ) ∆. We ′ denote this tensor product of representations⊗ also by Dj Dj . ⊗ ◦ ⊗ In general, the tensor product of two irreducible representation is no longer irreducible. In fact, in complete analogy to the classical case we have an isomorphism of representations

′ ′ ′ ′ Dj Dj = D|j−j | D|j−j |+1 ... Dj+j (2.2) ⊗ ∼ ⊕ ⊕ ⊕ decomposing the tensor product into the Clebsch-Gordan series. This isomorphism, viewed as a transformation of basis

j, m = C (j , j , j m , m , m) j , m j , m (2.3) | i q 1 2 | 1 2 | 1 1i⊗| 2 2i j ,j ,m ,m 1 2X1 2 1There is a second series of irreducible representations with negative eigenvalues of K, which we will not take into account, since they have no undeformed limit. 2.1 Representation Theory of the q-Lorentz Algebra 19 deﬁnes the q-Clebsch-Gordan coeﬃcients, which can be calculated in a closed form (Sec. A.1.1). The two most important cases are the construction of a scalar and the construction of a vector out of two vector representations, where the right hand side of Eq. (2.3) may be viewed as the scalar and the vector product of two 3-vectors.

2.1.2 Clebsch-Gordan Coeﬃcients of the q-Lorentz Algebra

As an algebra the q-Lorentz algebra is the tensor product of two q(sl2). Hence, every finite irreducible representation is composed of two irreducibUle representations Dj1 and Dj2 of (sl ), that is, the vector space D(j1,j2) := Dj1 Dj2 with Uq 2 ⊗ the representation map ρ(j1,j2) := ρj1 ρj2 . Viewing the decomposition of the ⊗ q-Lorentz algebra into two (sl ) as chiral decomposition, we call Dj1 the left Uq 2 handed and Dj2 the right handed part of the representation. D(j1,j2) is not a -representation, since the -operation of the q-Lorentz algebra is not the tensor ∗ ∗ product of the ’s of q(sl2). Therefore, all finite irreducible representations are non-unitary. This∗ is aU sign of the non-compactness of the q-Lorentz algebra on a representation theoretic level. Next, we consider the tensor product of two representations. Again, its vector (j ,j ) (j′ ,j′ ) space is just the tensor product D 1 2 D 1 2 . The representation map is (j ,j ) (j′ ,j′ ) ⊗ again ρ = (ρ 1 2 ρ 1 2 ) ∆, where ∆ is now the coproduct of the q-Lorentz algebra as defined⊗ in Eq. (1.52).◦ The coproduct is calculated by, firstly, taking the (sl ) coproduct of the two (sl ) tensor factors, then interchanging the Uq 2 Uq 2 2. and 3. tensor factor, and, finally, conjugating with the universal -matrix in the 2. and 3. position of the 4-fold tensor product. Algebraically, tRhe last step is a complicated inner automorphism, since exists only as an infinite formal R power series. However, when we apply the representation maps, becomes a finite (j j′ ) (j j′ ) matrix R =(Rab ) R 2 1 × 2 1 cd (j ,j ) (j′ ,j′ ) j j′ ρ 1 2 ρ 1 2 ( )=1 (ρ 2 ρ 1 )( ) 1=:1 R 1 , (2.4) ⊗ R23 ⊗ ⊗ R ⊗ ⊗ ⊗ and the inner automorphism becomes a simple basis transformation. Putting things together, we see how to reduce the product of two q-Lorentz representations. Up to a change of basis we reduce the tensor product of the 1. with the 3. and the 2. with the 4. of the q(sl2)-subrepresentations, each by means of the Clebsch-Gordan series of (sl ).U Uq 2 ′ ′ (j1,j2) (j1,j2) (k1,k2) D D ∼= D (2.5) ⊗ ′ ′ |j1−j |≤k1≤j1+j 1′ M 1′ |j2−j2|≤k2≤j2+j2 Written out for the important case of the product of two vector representations, this formula becomes

( 1 , 1 ) ( 1 , 1 ) (0,0) (1,0) (0,1) (1,1) D 2 2 D 2 2 = D D D D , (2.6) ⊗ ∼ ⊕ ⊕ ⊕ 20 2. Structure of the q-Lorentz Algebra which corresponds to the decomposition of a 4 4 matrix viewed as a second rank Lorentz tensor into the scalar trace part, a left× and a right chiral 3-vector, and the traceless symmetric part (Sec. A.1.3). So far, the representation theory is in complete accordance with the undeformed case. New is the appearance of an R-matrix, which matters as soon as we want to write down the above isomorphisms explicitly. The matrix representing isomorphism 2.5 is the product of two Clebsch-Gordan coeﬃcients contracted with the R-matrix. Musing for a while about the right positions of the indices, we ﬁnd

(k ,k ), (n , n ) = C (j , j′ ,k m , b, n )C (j , j′ ,k a, m′ , n ) | 1 2 1 2 i q 1 1 1 | 1 1 q 2 2 2 | 2 2 −1 m m′ ′ ′ ′ ′ (XR ) 2 1 (j , j ), (m , m ) (j , j ), (m , m ) , (2.7) × ab | 1 2 1 2 i⊗| 1 2 1 2 i where we sum over repeated indices, and where the labeling of the free indices is the same as in Eq. (2.5). This deﬁnes the Clebsch-Gordan coeﬃcients of the q-Lorentz algebra

j j′ k m m′ n 1 1 1 1 1 1 := j j′ k m m′ n 2 2 2 2 2 2 q ′ ′ ′ −1 m m′ C (j , j ,k m , b, n )C (j , j ,k a, m , n )(R ) 2 1 . (2.8) q 1 1 1 | 1 1 q 2 2 2 | 2 2 ab Xa,b 2.2 Tensor Operators 2.2.1 Tensor Operators in Hopf Algebras Recall that there is a left and right action of any Hopf algebra H on itself given by

adL(g) ⊲ h := g(1)hS(g(2)) , h⊳ adR(g) := S(g(1))hg(2) (2.9) for g, h H, called the left and right Hopf adjoint action, respectively. In ∈ general, this action will be highly reducible. In fact, if a linearly independent set Aµ H of operators generates an invariant subspace D of the left Hopf adjoint action,{ ∈ this} induces a matrix representation map ρ of H by

µ′ adL(h) ⊲ Aµ = Aµ′ ρ(h) µ , (2.10) turning D into a representation. The set of operators Aµ with this property is called a left D-tensor operator of H, indicated by a lower{ index.} It will be called irreducible if D is irreducible. If in addition H is equipped with a -operation, we can demand that D is a -representation. ∗ There are other useful types∗ of tensor operators. If a set of operators Aµ { } transforms as

µ µ µ′ (adLh) ⊲ A = ρ(Sh) µ′ A , (2.11) 2.2 Tensor Operators 21 we will call it a left upper or congredient tensor operator, denoted by an upper index. Its transformation is congredient in the sense that

µ µ (adLh) ⊲ (AµB ) = [(adLh(1)) ⊲ Aµ][(adLh(2)) ⊲ B ] µ′ µ µ′′ µ = Aµ′ ρ(h(1)) µ ρ S(h(1)) µ′′ B = ε(h) AµB , (2.12)

µ µν that is, AµB is a scalar operator. If g is a metric for the representation under µν consideration and Aµ and Bν are left tensor operator then g AµBν is a scalar. This is true for the q-spinor metric εab, the metric gAB of vector representations µν of q(su2) and the q-Minkowski metric η , as deﬁned in Eqs. (1.10), (2.23), andU (3.16), respectively. We conclude, that the convention for the position of tensor operator indices is consistent with raising and lowering indices as usual, µ µµ′ A = g Aµ′ . Moreover, we conclude that

µµ′ ν′ µ g gν′ν ρ(h) µ′ = ρ(Sh) ν . (2.13)

If we deal with a Hopf- -algebra and ρ is a -representation, we can apply to Eq. (2.10) and get ∗ ∗ ∗

( S)(h )(A )∗(h )∗ = ((Sh)∗) (A )∗S ((Sh)∗) = (ad (Sh)∗) ⊲ (A )∗ ∗◦ (2) µ (1) (1) µ (2) L µ ∗ µ′ ∗ ∗ µ ∗ ∗ µ =(Aµ′ ) ρ(h) µ =(Aµ′ )ρ(h ) µ′ =(Aµ′ ) ρ(S[(Sh) ]) µ′ , (2.14) from which we deduce

∗ ∗ ∗ µ ∗ (adL(Sh) ) ⊲ (Aµ) = ρ(S[(Sh) ]) µ′ (Aµ′ ) . (2.15)

∗ Comparing this with Eq. (2.11), we conclude that (Aµ) is a congredient left tensor operator. Let us now consider tensor operators Aµ˜ with respect to the right Hopf-adjoint action

µ˜ µ˜ µ µ˜′ A ⊳ (adRh)= S(h(1))A h(2) = ρ(h) µ′ A , (2.16) which we call right upper tensor operators, distinguished from left upper tensor operators by putting a tilde over their indices. Let Aµ be a left upper tensor operator and let there be an extension of the antipode of H on Aµ, for example, Aµ might be an element of H. Then we can apply S to Eq. (2.11) and obtain

µ µ S(S(h(2)))S(A )S(h(1))= S((Sh)(1)))S(A )(Sh)(2)) µ µ µ′ = S(A ) ⊳ (adRSh)= ρ(Sh) µ′ S(A ) . (2.17)

Thus, S(Aµ) is a right upper tensor operator. Note, that within a Lie algebra we would have S(Aµ) = Aµ. Hence, in a Lie algebra a right tensor operator is the same as a left tensor− operator. This is 22 2. Structure of the q-Lorentz Algebra why in the undeformed case we need not distinguish between indices with and without a tilde.

Finally, we deﬁne a right lower tensor operator Aµ˜ to transform as S(Aµ), that is,

−1 µ′ Aµ˜ ⊳ (adRh)= Aµ˜′ ρ(S h) µ . (2.18)

ν˜ One can check that we can raise and lower indices as usual, Aµ˜ = gµνA , and µ˜ that A Bµ˜ is a scalar operator. Note that being a left or a right scalar is the same thing: A scalar is an operator that commutes with H.

2.2.2 Tensor Operators of Uq(su2)

0 Most tensor operators of q(su2) that we will consider are D -tensor opera- U 1 tors, which will be called q(sl2)-scalars, and D -tensor operators, called q(sl2)- vectors. One big advantageU of grouping several operators to a (sl )-tensorU Uq 2 operator lies in the q-Wigner-Eckart theorem:

λ Theorem 1. Let Aµ be a left D -tensor operator of q(su2) and let there be U j j′ a representation of q(su2) with irreducible subrepresentations D and D with bases j, m and Uj′, m′ . Then there exists a number j′ A j such that {| i} {| i} h k k i j′, m′ A j, m = C (λ, j, j′ µ, m, m′) j′ A j (2.19) h | µ| i q | h k k i for all m, m′. This number is called the reduced matrix element of the tensor operator Aµ [51].

If we have degeneracy of the j, m basis, the reduced matrix elements will depend on additional quantum numbers| i but not on m. Whenever a q-Clebsch- ′ ′ ′ Gordan coefficient Cq(λ, j, j µ, m, m ) vanishes for all m, m , the reduced matrix element is not defined uniquely.| In that case we set j′ A j := 0 for convenience. h k k i ′ Looking at the definition (2.9), we see that adL(g) ⊲ (hh ) = (adL(g(1)) ⊲ ′ ′ h)(adL(g(2)) ⊲ h ). Hence, the product of a D- and a D -tensor operator is a ′ D D -tensor operator. Just as for the representations of q(sl2) we have a Clebsch-Gordan⊗ decomposition of the product of tensor operatoU rs:

a b Proposition 2. Let Aα be a D -tensor operator and Bβ a D -tensor operator of (sl ). Then Uq 2 C := C (a, b, c α,β,γ)A B (2.20) γ q | α β Xα,β is a Dc-tensor operator of (sl ). Uq 2 2.2 Tensor Operators 23

If we now take the matrix elements of a tensor operator Cγ constructed in this way, we ﬁnd with the aid of the q-Wigner-Eckart theorem relations between the reduced matrix elements

′′ ′ ′′ ′′ ′ ′ j C j = Rq(a, b, j c, j , j ) j A j j B j . (2.21) h k k i ′ | h k k ih k k i Xj

Here Rq denote the q-Racah coeﬃcients deﬁned by the expression

R (a, b, j c, j′, j′′) := C (c, j, j′′ γ, m, m′′)−1 q | q | ′′ ′ ′′ ′ ′ Cq(a, b, c α,β,γ)Cq(a, j, j α, m , m )Cq(b, j, j β, m, m ) , (2.22) × ′ | | | α,β,mX ′′ which can be proven not to depend on m, m as the arguments of Rq indicate. Values of the q-Racah coeﬃcients are given in Sec. A.1.1. The two cases of Eq. (2.20) that we encounter most frequently are the construction of a scalar and the construction of a vector operator out of two vector operators. This suggests the deﬁnition

AB AB [4] g := [3]Cq(1, 1, 0 A, B, 0) , ε C = Cq(1, 1, 1 A,B,C) , (2.23) − | −s[2] | p where the capital Roman indices run through 1, 0, 1 = , 3, + . Values are given in Sec. A.1.2. Proposition 2 tells us that we{− can deﬁne} {− a scalar} and a vector product of two vector operators XA and YA by

X~ Y~ := X Y gAB , (X~ Y~ ) := iX Y εAB , (2.24) · A B × C A B C where the imaginary unit is needed for the right undeformed limit.2 By deﬁnition, the scalar product is a scalar and the vector product is a vector operator in the sense of Eq. (2.10).

2.2.3 The Vector Form of Uq(su2) For a set of operators to be interpreted as q-angular momentum, it will have to generate the symmetry of rotations on the one hand, but on the other hand it will itself have to transform like a vector under rotations. In other words, this set must be a vector operator generating (su ). In the EFK-form of (sl ) it Uq 2 Uq 2 is not obvious, what this vector operator could be. We begin our search for such a vector operator by giving the explicit conditions for A to be a irreducible Dj-tensor operator of (sl ): Inserting Eqs. (2.1) in µ Uq 2 2See Sec. A.1.2, in particular the remark above Eq. (A.16). 24 2. Structure of the q-Lorentz Algebra

Eq. (2.10) we get

EA A E = q(µ+1) [j + µ + 1][j µ] A K µ − µ − µ+1 −2µ −µ F Aµ q AµF = q p[j + µ][j µ + 1] Aµ−1 (2.25) − 2µ − KAµ = q ApµK.

To find a vector operator in q(su2) satisfying these conditions we first look for a highest weight vector J andU let (sl ) act on it by the left Hopf-adjoint action, + Uq 2 giving us the subrepresentation generated by J+. The condition adL(E) ⊲J+ =0 for J+ to be a highest weight vector is equivalent to [E,J+] = 0. Thus, J+ must be in the centralizer of E, a very restrictive condition most obviously satisfied by E itself. The results of the Hopf-adjoint action of the ladder operators E and F on E are −1 2 adL(F ) ⊲ E = K (KF E EKF ) , adL(F ) ⊲ E = [2]KF 3 − − , adL(F ) ⊲ E =0 , adL(EF ) ⊲ E = [2]E (2.26) ad (EF 2) ⊲ E = [2]K−1(KF E EKF ) , L − which shows that we can indeed interpret E as a highest weight vector of a vector representation. Comparing the Hopf-adjoint action with the vector representation as given in Eqs. (2.1), one finds that

− 1 J− := q[2] 2 KF J := q[2]−1K−1(KF E EKF ) = [2]−1(q−1EF qF E) (2.27) 3 − − − − 1 J := [2] 2 E + − form a vector operator.3 How can we describe the subalgebra of q(su2) generated by JA? After some calculations we ﬁnd that the commutation relationsU of the J’s do not close. Since the commutation relations (2.10) are given by the adjoint action of the set of generators on itself, this is due to the fact that coproduct and antipode of the J’s cannot be expressed by J’s again. We can help ourselves out by introducing the additional generator W := K λJ = K λ[2]−1(q−1EF qF E) , (2.28) − 3 − − so the commutation relations can be written as J J εAB = WJ , J W = WJ , W 2 λ2J J gAB =1 , (2.29) A B C C A A − A B where the last equation expresses that W and the J’s are not algebraically independent. The -structure reads on the generators ∗ J ∗ = qJ , J ∗ = J , W ∗ = W, (2.30) + − − 3 3 3 A −3 Elsewhere [43], the vector generators have been deﬁned as L = q JA. − 2.3 The q-Lorentz Algebra as Quantum Double 25

∗ A that is, (JA) = J . We will call the subalgebra of q(sl2) generated by JA, W with relations (2.29) and -structure (2.30) the vectorialU form of (su ). Note ∗ Uq 2 that the vectorial form of q(sl2) is a proper subalgebra of q(sl2) since it does not contain K−1. We do needUK−1 to write down the Hopf structure:U the coproduct ∆(J )= J K +1 J ± ± ⊗ ⊗ ± ∆(J )= J K + K−1 J + λ(qK−1J J + q−1K−1J J ) (2.31) 3 3 ⊗ ⊗ 3 + ⊗ − − ⊗ + ∆(W )= W K λK−1 J λ2(qK−1J J + q−1K−1J J ) , ⊗ − ⊗ 3 − + ⊗ − − ⊗ + the antipode S(J )= J K−1 ± − ± S(J )= J λ−1(K K−1) (2.32) 3 3 − − S(W )= W, and the counit ε(JA) = 0, ε(W ) = 1.

2.3 The q-Lorentz Algebra as Quantum Double op 2.3.1 Rotations and the SUq(2) Algebra of Boosts In Sec. 1.2.2 the commutation relations of the q-Lorentz group have been chosen to preserve an SUq(2) substructure, physically interpreted as rotations. That is, the multiplication of the two copies of SL (2) is a Hopf- -homomorphism q ∗ projecting the q-Lorentz group onto SUq(2). On the quantum algebra level, the dual of multiplication is comultiplication. Hence, the mapping

i : (su ) ∆ (sl ) (sl )= (sl (C)) (2.33) Uq 2 −→ Uq 2 ⊗ Uq 2 Uq 2 ought to define a (su ) Hopf- -subalgebra of the q-Lorentz algebra. Uq 2 ∗ Given the properties of the coproduct, it is obvious that i is an algebra homomorphism. It is less clear, whether i preserves the Hopf structure and the -structure of (su ). For the coproducts we find ∗ Uq 2 −1 (∆ C i)(h)= (h h h h ) = h h h h Uq (sl2( )) ◦ R23 (1) ⊗ (3) ⊗ (2) ⊗ (4) R23 (1) ⊗ (2) ⊗ (3) ⊗ (4) = (i i) ∆ (h) , (2.34) ⊗ ◦ Uq (sl2) which shows that i is a coalgebra map. In the same manner we find that i preserves the counit (trivial), the antipode

−1 −1 (S C i)(h)= S(h ) S(h ) = (Sh) (Sh) Uq(sl2( )) ◦ R21 (1) ⊗ (2) R21 R21 (2) ⊗ (1) R21 =(Sh) (Sh) =(i S )(h) , (2.35) (1) ⊗ (2) ◦ Uq(sl2) and the -structure ∗ (i(h))∗ = (h )∗ (h )∗ −1 = (h∗) (h∗) −1 R21 (2) ⊗ (1) R21 R21 (2) ⊗ (1) R21 =(h∗) (h∗) = i(h∗) . (2.36) (1) ⊗ (2) 26 2. Structure of the q-Lorentz Algebra

We conclude that i( q(su2)) is indeed a q(su2) Hopf- subalgebra of the q- Lorentz algebra.4 SinceU in the undeformedU case the embedding∗ of the rotations in the Lorentz algebra is given by the coproduct, too, i( q(su2)) has the right undeformed limit. This strongly suggests to interpret i( U(su )) as the quantum Uq 2 subsymmetry of physical rotations.

There is another Hopf- subalgebra of q(sl2(C)). Let , denote the dual pairing of (sl ) and SL∗(2) as deﬁnedU in Sec. 1.3.1. Weh deﬁnei a map j : Uq 2 q SU (2) (sl (C)) by q → Uq 2 j(h) := −1 , h , (2.37) hR31 R23 3i where the subscripts denote the position in the tensor product, h3 := 1 1 h, and where the dual pairing acts only on the third tensor factor. Let⊗ us show⊗ some properties of this map. we have

j(gh)= −1 ,g h = ∆ ( −1 ),g h = −1 −1 ,g h hR31 R23 3 3i h 3 R31 R23 3 4i hR41 R31 R24R23 3 4i = −1 −1 ,g h = j(h)j(g) , (2.38) hR41 R24R31 R23 3 4i telling us that j is an algebra anti-homomorphism. Next we consider the coproduct

−1 −1 −1 −1 (∆ C j)(h)= ∆ id ( ), h = R , h Uq(sl2( )) ◦ h ⊗ R31 R23 3i hR23 51 R53 R25R45R23 5i = −1 −1 , h = (j j) ∆ (h) , (2.39) hR51 R25R53 R45 5i ⊗ ◦ SUq(2) so j is a coalgebra homomorphism, too. The calculation for the counit is trivial. op So far we can say that j is a bialgebra homomorphism from SUq(2) to the op q-Lorentz algebra. SUq(2) becomes a Hopf algebra, when we equip it with a antipode and -structure according to ∗ Sop := S−1 , op := S2 , (2.40) ∗ ∗◦ where S is the usual antipode of SUq(2). Let us check now if j preserves this Hopf structure as well. We begin with the antipode

op −1 −1 −1 −1 −1 (j S )(h)= 31 23,S (h3) = 23 31, h3 = 21 31 23 21 , h3 ◦ hR R i −hR1 R −1 i hR R R R i = 21[(S S id)( 31 23)] 21 , h3 =(SUq(sl2(C)) j)(h) , hR ⊗ ⊗ R R R i ◦ (2.41)

4 It is the appearance of the -matrices in the Hopf structure of q(sl2(C)), which ensures R U the compliance of the embedding i with the Hopf structures. This is why q(so4) does not possess a Hopf subalgebra of rotations. U 2.3 The q-Lorentz Algebra as Quantum Double 27 which is indeed preserved. Finally, we have the -structure ∗ op j h∗ = −1 , (S2h )∗ = [id id (S2 S)]( −1 ), h hR31 R23 3 i h ⊗ ⊗ ◦∗◦ R31 R23 3i −1 −1 −1 = [id id ( S )]( 31 23), h3 = ( id)( 13 32 ), h3 h ⊗ ⊗ ∗◦ R R i h ∗⊗∗⊗ R R i = ( id)( −1 −1 ), h h ∗⊗∗⊗ R12 R32 R13R12 3i = [( id)( −1 )] −1, h = −1 , h (∗⊗∗) −1 hR21 ∗⊗∗⊗ R32 R13 R21 3i R21 hR32 R13 3i R21 = j(h) ∗ . (2.42)

op We conclude thatj is a Hopf- algebra homomorphism from SUq(2) to the q- ∗ op Lorentz algebra. Hence, j(SUq(2) ) is indeed a Hopf- subalgebra of q(sl2(C)). We will call it the subalgebra of the boosts. ∗ U

2.3.2 L-Matrices and the Explicit Form of the Boost Algebra To calculate the explicit form of the algebra of boosts we introduce the com- putational tool of L-Matrices [16]. Let ρj be the representation map of the Dj-representation of (sl ). We deﬁne matrices of generators by applying ρj to Uq 2 one tensor factor of the universal -matrix = , R R R[1] ⊗ R[2] (Lj )a := ρj( )a , (Lj )a := ρj( −1)a −1 . (2.43) + b R[1] R[2] b − b R[1] b R[2] 1 Here, we need the L-matrices for j = 2 , where we get

1 1 1 1 1 − 2 − 2 − 2 1 2 2 K q λK E 2 K 0 L+ = 1 , L− = 1 1 − 1 (2.44) 0 K 2 q 2 λFK 2 K 2 − with respect to the basis , + . The appearance of the square roots of K comes {− } from the fact that only exists as formal power series. We can derive someR properties of the L-matrices from the properties of : Ap- plying id ρj ρj to the quantum Yang-Baxter equation = R ⊗ ⊗ R12R13R23 R23R13R12 we obtain

j j ′ ′ j ′ j ′ a ′ d ′ c d ab ′ ′ b a (L+) c (L+) d R cd = R a b (L+) d(L+) c (2.45) and in an analogous manner

j j ′ ′ j ′ j ′ a ′ b ′ c d ab ′ ′ b a (L−) c (L−) d R cd = R a b (L−) d(L−) c j j ′ ′ j ′ j ′ (2.46) a ′ b ′ c d ab ′ ′ b a (L−) c (L+) d R cd = R a b (L+) d(L−) c .

From the coproduct properties (∆ id)( )= , (id ∆)( −1)= −1 −1 ⊗ R R13R23 ⊗ R R12 R13 and from (ε id)( ) = 1 = (id ε)( −1) it follows that ⊗ R ⊗ R ∆ (Lj )a =(Lj )a (Lj )b , ε (Lj )a = δa . (2.47) ± c ± b ⊗ ± c ± b b 28 2. Structure of the q-Lorentz Algebra

j −1 −1 −1 Finally, we apply id ρ id to the form 13 23 12 13 = 12 23 of the Yang-Baxter equation⊗ in order⊗ to get R R R R R R

−1 (Lj )b (Lj )a =(Lj )a (Lj )b . (2.48) R + c ⊗ − b R + b ⊗ − c Now, we can compute the explicit form of the boosts. Observing that the dual a pairing of SUq(2) and q(su2) (Sec. 1.3.1) can be expressed on the matrix M b U a 1 a of generators of SU (2) by h, M = ρ 2 (h) , we get for the boost generators q h bi b

a a −1 a −1 1 −1 a B := j(M )= , 1 1 M =( ′ )ρ 2 ( ′ ) c c hR31 R23 ⊗ ⊗ ci R[2] ⊗ R[1 ] R[1] R[2 ] c 1 1 −1 1 −1 a 1 b a b = ρ 2 ( ) ′ ρ 2 ( ′ ) = L 2 L 2 , (2.49) R[2] R[1] b ⊗ R[1 ] R[2 ] c − b ⊗ + c explicitly,

1 − 1 − 1 1 − 1 a K 2 K 2 q 2 λK 2 K 2 E a b B b = 1 ⊗ 1 − 1 − 1 1 ⊗2 1 − 1 =: . q 2 λFK 2 K 2 K 2 K 2 λ FK 2 K 2 E c d − ⊗ ⊗ − ⊗ (2.50)

The commutation relations are

ba = qab, ca = qac, db = qbd, dc = qcd (2.51) bc = cb, da ad =(q q−1)bc, da qbc =1 . − − −

a a b The coproduct, ∆(B c) = B b B c, is the same as for SUq(2) just as the a a ⊗ op −1 counit, ε(B b)= δb . For the antipode we had S = S and for the -structure op 2 a ∗ b a ∗op ∗op b := S . Since (M b) = S(M a), it follows that (M b) = S (M a) in ∗ op∗◦ a ∗ b SUq(2) and, consequently, the unitarity condition (B b) = S(B a) holds in (sl (C)) as well. Written out this is Uq 2 a b d qb a b ∗ d q−1c S = − , = − . (2.52) c d q−1c a c d qb a − −

a op If we want to verify that the B b are the generators of a SUq(2) subalgebra using the deﬁnition of the q-Lorentz algebra only, we ﬁnd that this is extremely tedious.

2.3.3 Commutation Relations between Boosts and Rotations

Now, we have to ﬁgure out the commutation relations between rotations and boost, embedded into (sl (C)) by the maps i and j, respectively. For l (su ) Uq 2 ∈ Uq 2 2.3 The q-Lorentz Algebra as Quantum Double 29 and h SU (2)op the embedding is ∈ q −1 j(h)i(l)= 31 23, h3 (l(1) l(2)) hR−1 R i ⊗ −1 = l ′ l ′ l S(l ), h R[2] (1) ⊗ R[1 ] (2) hR[1] R[2 ] (3) (4) i −1 −1 = l l ′ l ′ S(l ), h R[2] (1) ⊗ (3)R[1 ] hR[1] (2)R[2 ] (4) i −1 −1 = l l ′ l ′ S(l ), h (2)R[2] ⊗ (3)R[1 ] h (1)R[1] R[2 ] (4) i −1 −1 = l l ′ l , h ′ , h S(l ), h (2)R[2] ⊗ (3)R[1 ] h (1) (1)ihR[1] R[2 ] (2)ih (4) (3)i = l , h i(l )j(h ) S(l ), h . (2.53) h (1) (1)i (2) (2) h (3) (3)i The commutation relations which can be read oﬀ this equation are precisely the ones of the quantum double [12, 13]. For the generators they write out

a a 1 −1 a′ 1 a a′ 1 −1 b′ a 1 −1 a′ B E = EB ′ ρ 2 (K ) + Kρ 2 (E) ′ B ′ ρ 2 (K ) B ′ ρ 2 (EK ) b a b a b b − a b a 1 −1 a′ a′ −1 1 −1 a a′ 1 b′ 1 a a′ B F = F ρ 2 (K ) ′ B K ρ 2 (K ) ′ B ′ ρ 2 (KF ) + ρ 2 (F ) ′ B b a b − a b b a b a 1 a a′ 1 −1 b′ B bK = ρ 2 (K) a′ B b′ ρ 2 (K ) b . (2.54) Explicitly, this gives us

3 −1 a b qEa q 2 b q Eb E = 3− 3 −1 − 1 c d qEc + q 2 Ka q 2 d q Ed + q 2 Kb − − 1 − 1 −1 − 1 a b qF a + q 2 c qFb q 2 K a + q 2 d (2.55) F = −1 −−1 − 5 −1 c d q F c q Fd q 2 K c − a b a q−2b a b a q2b K = K , K−1 = K−1 . c d q2c d c d q−2c d We summarize: op Deﬁnition 4. The Hopf- algebra generated by SUq(2) and q(su2) with cross commutation relations ∗ U hl = l , h l h S(l ), h (2.56) h (1) (1)i (2) (2) h (3) (3)i or, equivalently, lh = S(l ), h h l l , h (2.57) h (1) (1)i (2) (2) h (3) (3)i op for h SUq(2) and l q(su2), is the quantum double form of the q-Lorentz algebra∈ [21]. ∈ U op Finally, if we want to invert the embedding i j : q(su2) SUq(2) (sl (C)) we ﬁnd ⊗ U ⊗ → Uq 2 − 1 3 −1 1 −1 −1 E 1= qK 2 (Ea q 2 λ b) , 1 E = q 2 λ a b ⊗ − ⊗ − 1 −1 −1 1 − 1 −1 F 1= q 2 λ ca , 1 F = qK 2 (F a + q 2 λ c) (2.58) ⊗ − ⊗ 1 1 −1 K 1= K 2 a , 1 K = K 2 a ⊗ ⊗ 30 2. Structure of the q-Lorentz Algebra

−1 op For these expressions to make sense we had to add the generator a to SUq(2) ± 1 and K 2 to (su ). From the viewpoint of representation theory this modiﬁca- Uq 2 tion seems to be insigniﬁcant.

2.4 The Vectorial Form of the q-Lorentz Algebra 2.4.1 Tensor Operators of the q-Lorentz Algebra The deﬁnition of tensor operators in Eq. (2.10) has been general. We just have to work it out for the q-Lorentz algebra. We begin by calculating for g h (sl (C)) ⊗ ∈ Uq 2 −1 −1 (g h) S (g h) =(g h ′ ) S( g ′ h ) ⊗ (1) ⊗ ⊗ (2) (1) ⊗ R[1] (1)R[1 ] ⊗ R[2] (2)R[2 ] ⊗ (2) −1 −1 −1 −1 =(g h ′ ) ( ′′ S( g ′ ) ′′′ ′′ S(h ) ′′′ ) (1) ⊗ R[1] (1)R[1 ]⊗ R[2 ] R[2] (2)R[2 ] R[2 ] ⊗ R[1 ] (2) R[1 ] −1 −1 =(g h ′ ) ( ′′ S( ′ )S(g ) ′′′ ′′ S(h ) ′′′ ) , (1) ⊗ R[1] (1)R[1 ] ⊗ R[2 ] R[2 ] (2) R[2]R[2 ] ⊗ R[1 ] (2) R[1 ] (2.59) where in the last step we have used that (id S)( −1) = . Hence, for T = An Bn (no summation of µ and ⊗ν) toR be a D(i,jR)-tensor opera- µν n µν ⊗ µν tor of q(sl2(C)) UP i µ′ j ν′ T ′ ′ ρ (g) ρ (h) = ad (g h) ⊲ (T ) µ ν µ ν L ⊗ µν n −1 n −1 = g A ′′ S( ′ )S(g ) ′′′ h ′ B ′′ S(h ) ′′′ (1) µνR[2 ] R[2 ] (2) R[2]R[2 ] ⊗ R[1] (1)R[1 ] µνR[1 ] (2) R[1 ] n X (2.60) must hold for all g h (sl (C)). ⊗ ∈ Uq 2 Some tensor operators of q(sl2(C)) can be derived from tensor operators of j U (j,0) q(su2): If Aµ is a D -tensor operator of q(su2) then Aµ 1 is a D -tensor Uoperator. We check this by inserting T =UA 1 in the last⊗ equation: µν µ ⊗ adL(g h) ⊲ (Aµ 1) ⊗ ⊗ −1 −1 −1 −1 = g A S( ′ ′′ )S(g ) ′′′ h ( ′ ′′ )S(h ) ′′′ (1) µ R[2 ]R[2 ] (2) R[2]R[2 ] ⊗ R[1] (1) R[1 ]R[1 ] (2) R[1 ] −1 −1 = g A S(g ) ′′′ h S(h ) ′′′ = g A S(g ) ε(h) (1) µ (2) R[2]R[2 ] ⊗ R[1] (1) (2) R[1 ] (1) µ (2) ⊗ j µ′ 0 =(A ′ 1) ρ (g) ρ (h) . (2.61) µ ⊗ µ In the same manner we verify that (1 A ) −1 is a D(0,j)-tensor operator: R21 ⊗ µ R21 −1 adL(g h) ⊲ 21(1 Aµ) 21 ⊗ R ⊗ R −1 −1 = g ′′ S( ′ )S(g ) ′′′ h ′ ′′ A S(h ) ′′′ (1)R[2 ] R[2 ] (2) R[2]R[2 ] ⊗ R[1] (1)R[1 ]R[1 ] µ (2) R[1 ] −1 −1 = g S(g ) ′′′ h A S(h ) ′′′ (1) (2) R[2]R[2 ] ⊗ R[1] (1) µ (2) R[1 ] = (1 h A S(h )) −1ε(g) R21 ⊗ (1) µ (2) R21 −1 j µ′ = (1 A ′ ) ε(g)ρ (h) . (2.62) R21 ⊗ µ R21 µ 2.4 The Vectorial Form of the q-Lorentz Algebra 31

2.4.2 The Vectorial Generators Now, it is obvious how we can define vectorial generators of the q-Lorentz algebra. Let J be the vector generator of (su ) as defined in Eqs. (2.27). We define5 A Uq 2 S := J 1 , R := (1 J ) −1 . (2.63) A A ⊗ A R21 ⊗ A R21 (1,0) (0,1) From the last section it is obvious that SA is a D -tensor and RA is a D - tensor operator, that is, a left and right chiral vector operator, respectively. More- (1,0) over, both RA and SA are vector operators with respect to rotations since D and D(0,1) induce a D1 vector representation of the (su ) subalgebra. Uq 2 We can raise the indices with the 3-metric of q(su2) introduced in Eq. (2.23), A AB U S = g SB, giving us a congredient vector operator,

A AA′ ad (g h) ⊲S = ad (g h) ⊲ (J ′ 1)g L ⊗ L ⊗ A ⊗ B AA′ j B′ =(J 1)g g ′ ρ (g) ′ ε(h) ⊗ B B A B j A = S ρ (Sg) B ε(Sh) , (2.64) and the same for R . By looking at the deﬁnition of the -structure of (sl (C)) A ∗ Uq 2 we immediately see that

∗ A (RA) = S . (2.65)

For the commutation relations of the algebra generated by RA and SA to close we yet have to embed the Casimir operator W of the vectorial form of q(su2), as deﬁned in Eq. (2.28), in the q-Lorentz algebra, that is6 U

V := W 1 , U := (1 W ) −1 =1 W. (2.66) ⊗ R21 ⊗ R21 ⊗ By construction the commutation relations of the R’s and U among each other are the same as for the L’s and W as given in Eqs. (2.29). The same holds for the S’s and V since these generators are embedded by an inner automorphism. To calculate the commutation relations of RA with SB we ﬁrst note that commuting with 1 J shows us that R21 ⊗ A 1 A′ R = J ′ ρ ( ) . (2.67) A R[2] ⊗ A R[1] A

Then we commute this expression with SA

1 A′ R S = J J ′ ρ ( ) A B R[2] B ⊗ A R[1] A 1 B′ 1 A′ = J ′ ρ ( ) ′ J ′ ρ ( ′ ) B R[2] BR[2 ] ⊗ A R[1 ]R[1] A 1 A′ 1 B′ = S ′ R ′ ρ ( ) ρ ( ) . (2.68) B A R[1] A R[2] B 5The operators R and S deﬁned here correspond to the operators q2[2]R and q2[2]S of [44]. 6The operator V deﬁned here corresponds to U ′ in [44]. − 32 2. Structure of the q-Lorentz Algebra

The representation of the universal -matrix appearing on the last line is propor- R tional to the R-matrix of SOq(3), deﬁned in Eq. (A.56). The RS-commutation relations can now be written as

′ ′ 2 ′ ′ A B RASB = q SB RA Rso3 AB , (2.69) where Rso3 is given explicitly in Eq. (A.58). We summarize

Deﬁnition 5. The algebra generated by RA, U, SA, V , where A runs through , +, 3 , with relations {− } R R εAB = UR , R U = UR , U 2 λ2gABR R = 1 (2.70a) A B C C A A − A B S S εAB = VS , S V = VS , V 2 λ2gABS S = 1 (2.70b) A B C C A A − A B

R S = q2S R q−1λg (gABS R )+ ε X εAB S R (2.70c) C D C D − CD A B C D X A B RAV = V RA , UV = VU, SAU = USA (2.70d) and -structure ∗ ∗ AB ∗ RA = g SB , U = V (2.70e) is called the vectorial or RS-form of the q-Lorentz algebra [44].

2.4.3 Relations with the other Generators

Let us ﬁrst express the vectorial generators RA and SA by the original generators of (sl (C)). For S and V the case is simple. We merely have to look up the Uq 2 A expressions for JA and W in Eqs. (2.27) and (2.28). For completeness we write them down once more

− 1 S− := q[2] 2 KF 1 −1 −1 ⊗ S3 := [2] (q EF qF E) 1 − ⊗ (2.71) − 1 S := [2] 2 E 1 + − ⊗ V := [K λ[2]−1(q−1EF qF E)] 1 . − − ⊗ For RA one might at ﬁrst sight expect formal power series, but as we have shown in the preceding section

1 A′ 1 A′ R = L ′ ρ ( ) = S (L ) J ′ . (2.72) A R[2] ⊗ A R[1] A − A ⊗ A 1 We only have to sum over the L−-matrix of q(su 2), which has been computed in Eq. (A.62) where we get U K−1 0 0 1 A 1 S (L−) B = λ[2] 2 F 1 0 (2.73)  2 2 2 1  q λ KF qλ[2] 2 KF K   2.4 The Vectorial Form of the q-Lorentz Algebra 33 with respect to the 1, 0, 1 = , 3, + basis, so the expressions for the R’s become {− } {− }

− 1 −1 − 1 −1 R = q[2] 2 K KF + λ[2] 2 F (q EF qF E) − ⊗ ⊗ − 2 2 − 1 2 q λ [2] 2 KF E − ⊗ R =1 [2]−1(q−1EF qF E) qλKF E (2.74) 3 ⊗ − − ⊗ − 1 R = [2] 2 K E + − ⊗ U =1 [K λ[2]−1(q−1EF qF E)] . ⊗ − − Next, let us express RA and SA by the generators of the quantum double form of the q-Lorentz algebra. For SA we ﬁnd

− 1 −1 − 1 1 S = q 2 λ [2] 2 K 2 c − − − 3 −1 −1 − 1 S = q 2 λ [2] K 2 (qcE Ec) 3 − −1 −1 − 1 − 1 = λ [2] K 2 (q 2 λEc + qKa qd) (2.75) − − 1 − 1 3 −1 S = q[2] 2 K 2 (q 2 λ b Ea) + − −1 − 1 −1 − 1 V = [2] K 2 (q Ka q 2 λEc + qd) . − ∗ To compute the corresponding expressions for RA we remember that S− = −1 ∗ ∗ q R+, S3 = R3, and S+ = qR−. With the -structure of rotations and −boosts as given in Eqs. (2.30) and− (2.52) this yields∗

− 1 − 1 − 5 −1 R− = [2] 2 K 2 (Fd + q 2 λ c) − 1 −1 −1 1 3 R = q 2 λ [2] K 2 (bF q F b) 3 − −1 −1 1 3 −1 −1 −1 = λ [2] K 2 ( q 2 λF b q K a + q d) (2.76) − − 1 −1 − 1 1 R = q 2 λ [2] 2 K 2 b + − −1 1 3 −1 −1 U = [2] K 2 (q 2 λF b + q K a + qd) . We also want to express the generators of boosts and rotations within the RS- algebra. For the vectorial generators of the rotations we ﬁnd [31, 43]

AB JC = V RC + USC + qλRASB ε C 2 2 AB (2.77) W = UV + q λ g RASB .

−1 While this yields an expression of K = W + λJ3, K is not a member of the − 1 RS-algebra proper. We must add K 2 by hand to the RS-algebra to write down expressions for the boosts

− 1 − 1 1 − 1 a = K 2 (V + λS ) , b = q 2 λ[2] 2 K 2 R (2.78a) 3 − + 1 1 − 1 − 1 c = q 2 λ[2] 2 K 2 S , d = K 2 (U + λR ) . (2.78b) − − 3 Chapter 3 Algebraic Structure of the q-Poincar´eAlgebra

3.1 The q-Poincar´eAlgebra 3.1.1 Construction of the q-Minkowski-Space Algebra As in the undeformed case, we want to construct the coordinate functions of

Minkowski space to form a matrix Xab˙ with a lower undotted and dotted index. For the cotransformations to be compliant with the -structure the has to act ∗ ∗ ∗ on Xab˙ as on a product φaψb˙ , that is, (Xab˙ ) := Xba˙ . For our purposes it is more b˙ convenient to work with the index structure Xa ,

b˙ A B b˙ ∗ qD B Xa := (Xa ) = − , (3.1) CD C q−1A − with respect to the , + basis. With this index structure the cotransformation {− } is1

b˙ b˙′ a′ b′ ρ (X )= X ′ (M M ) . (3.2) R a a ⊗ a ⊗ b ( 1 , 1 ) Upon dualizing, this right coaction of the q-Lorentz group becomes a left D 2 2 action of the q-Lorentz algebra. b˙ We want to construct the space algebra out of the algebra C Xa freely b˙ h i generated by the generators Xa divided by some relations. The generators have the dimension of a length, so we need homogeneous relations,2 which for the correct undeformed limit have to be of second order. We demand the resulting quotient algebra to be a q-Lorentz module algebra. This last requirement means that the quadratic terms that will be set zero must be the basis of a q-Lorentz submodule. For only if we divide the free module b˙ C Xa by an ideal generated by a submodule, the quotient will be a module h i b˙ d˙ ( 1 , 1 ) ( 1 , 1 ) again. The vector space generated by Xa Xc naturally forms a D 2 2 D 2 2 representation of the q-Lorentz algebra. By the Clebsch-Gordan-Series (2.6)⊗ this representation has the same four subrepresentations as in the undeformed case. To obtain the correct undeformed limit where the space-functions commute, it is 1Recall, that we think of the dot as belonging to X rather than to the index itself. 2For inhomogeneous relations we would need to introduce an additional dimensional parameter. 3.1 The q-PoincaréAlgebra 35 the submodules D(1,0) and D(0,1) that have to be set zero. The bases of those two submodules as computed in Eqs. (A.23) and (A.24) yield the relations 0= qBA q−1AB − 0= DA AD + λBB + BC q−2CB − − 0= DC CD + λDB − (3.3) 0= CA AC + λBA − 0= DA AD + λBB + CB q−2BC − − 0= qDB q−1BD, − which can be written more compactly as AB = q2BA,BD = q2DB, BC = CB AC CA = λBA , CD DC = λDB (3.4) − − AD DA = λB(B + q−1C) . − Now we can give the definition of the q-Minkowski-Space Algebra. Definition 6. The -algebra generated by A, B, C, D with -structure as in Eq. (3.1) and commutation∗ relations (3.4) is{ called the q-Minkowski-Space} ∗ algebra [20]. Mq The basis vector of the D(0,0) submodule yields a q-Lorentz scalar, that corresponds to the invariant quadratic length, X2, of Minkowski space. Up to nor- malization we get from Eq. (A.25) X2 := [2]−1(qDA + q−1AD q−1BC q−1CB q−1λBB) , (3.5) − − − which can be simplified with the commutation relations (3.4) to X2 = DA q−2BC. (3.6) − It turns out that this expression commutes with all generators of . Hence, it Mq can be viewed as the length Casimir of q-Minkowski space or, within a momentum representation, as mass Casimir of the q-Poincaréalgebra.

3.1.2 4-Vectors and the q-Pauli Matrices

We have constructed the q-Lorentz algebra to possess a q(su2) Hopf- subalge- U ∗ b˙ bra, viewed as the algebra of rotations. Hence, we are able to write Xa in a manifest 4-vector form, that is, split up its 4 degrees of freedom with respect to rotations into a scalar and a 3-vector. ( 1 , 1 ) The D 2 2 representation induces a representation on the subalgebra of rotations. To compute the representation map ρ of the latter we have to embed ( 1 , 1 ) (su ) with i = ∆ and then apply the representation map ρ 2 2 yielding Uq 2 1 1 ρ =(ρ 2 ρ 2 ) ∆ . (3.7) ⊗ ◦ 36 3. Algebraic Structure of the q-Poincar´eAlgebra

In other words, this induced representation is simply the tensor representation 1 1 D 2 D 2 which reduces according to the Clebsch-Gordan series ⊗ 1 1 0 3 D 2 D 2 = D D (3.8) ⊗ ∼ ⊕ to the direct sum of a scalar and a vector representation. Explicitly, this reduction b˙ of Xa into a 4-vector is expressed by the q-Clebsch-Gordan coeﬃcients,

−1 − 1 1 1 b˙ − 1 1 1 b˙ X = q [2] 2 C ( , , 0 a, b, 0)X , X = [2] 2 C ( , , 1 a, b, C)X , (3.9) 0 q 2 2 | a C q 2 2 | a where C runs through ( 1, 0, 1) = ( , 3, +) and we sum over repeated indices. − 1 − − The factor [2] 2 has been introduced to ensure the right undeformed limit, the −1 factor of q in the deﬁnition of X0 is traditional [43]. Written out, we get

−1 −1 1 − 1 X = q [2] (q 2 C q 2 B) 0 − − 1 X− = [2] 2 A (3.10) − 1 X+ = [2] 2 D −1 − 1 1 X3 = [2] (q 2 C + q 2 B) . The back transformation is

1 1 A = [2] 2 X , B = q 2 (X X ) (3.11a) − 3 − 0 − 1 3 1 C = q 2 X3 + q 2 X0 ,D = [2] 2 X+ . (3.11b) Expressed in terms of the 4-vector generators, the commutation relations (3.4) become

X−X0 = X0X− , X+X0 = X0X+ , X3X0 = X0X3 q−1X X qX X = λX X , q−1X X qX X = λX X (3.12) − 3 − 3 − − − 0 3 + − + 3 − + 0 X X X X λX X = λX X − + − + − − 3 3 − 3 0 Using the q-deformed ε-tensor (2.23) this can be written more compactly as

X X = X X , X X εAB = λX X . (3.13) 0 A A 0 A B C − 0 C For the -structure we get ∗ ∗ ∗ A X0 = X0 , (XA) = X , (3.14) for the scalar product (3.5)

X2 = X2 + q−1X X + qX X X2 = X2 X X gAB . (3.15) 0 − + + − − 3 0 − A B 2 µν From this, we can read off the 4-metric, X = XµXνη , with η00 =1 , ηAB = gAB (3.16) − 3.1 The q-PoincaréAlgebra 37 and zero otherwise. We also could have computed the metric directly from the formulas of the Clebsch-Gordan coefficients. If we write the back transformation (3.11) as

b˙ b˙ Xa = Xµ(σµ)a , (3.17) µ X this deﬁnes the q-Pauli matrices

b˙ 1 1 1 b˙ 1 1 1 (σ ) = q[2] 2 C ( , , 0 a, b, 0) , (σ ) = [2] 2 C ( , , 1 a, b, C) . (3.18) 0 a q 2 2 | C a q 2 2 | For the usual index structure we have to lower the dotted index.

˙′ b ′ (σµ)ab˙ =(σµ)a εb b (3.19)

The q-Pauli matrices with lower undotted and dotted indices are

− 1 q 0 1 0 q 2 1 0 0 q 0 2 2 σ0 = , σ− = [2] , σ+ = [2] 1 , σ3 = − −1 0 q 0 0 q 2 0 0 q − (3.20) with respect to the basis , + . If we compare the q-Pauli matrices with the 1 {− } spin- 2 representation of JA we ﬁnd

1 −1 ρ 2 (JA) = [2] σA . (3.21)

This tells us that if we raise (and lower) the vector index of σA as usual by A AA′ † A σ := g σA′ we get (σA) = σ , that is,

A ab˙ (σA)ba˙ =(σ ) . (3.22)

From Eq. (2.29) we deduce

AB −1 σA σB ε C = [4][2] σC . (3.23)

Further relations which are not representations of relations within the algebra of rotations can be found by explicit calculations

A B BA C BA C σ σ = ε C σ + g , σAσB = σC εA B + gAB . (3.24)

b˙ The basis transformation from the matrix generators Xa to the 4-vector generators Xµ deﬁnes a matrix representation Λ of the q-Lorentz algebra by

µ′ (g h) ⊲X = X ′ Λ(g h) . (3.25) ⊗ µ µ ⊗ µ 38 3. Algebraic Structure of the q-Poincar´eAlgebra

Using the formulas for the basis transformation Eqs. (3.9) and (3.17) we get

˙ ˙ (g h) ⊲X =(g h) ⊲ q−2[2]−1(σ ) bX b ⊗ 0 ⊗ 0 a a −2 −1 b˙ 1 a′ 1 b′ b˙′ = q [2] (σ0)a ρ 2 (g) a ρ 2 (h)bXa′ −2 −1 b˙ 1 a′ 1 b′ b˙′ = q [2] (σ0)a ρ 2 (g) a ρ 2 (h) b(σµ)a′ Xµ = X Λ(g h)µ (3.26) µ ⊗ 0 and

(g h) ⊲X =(g h) ⊲ [2]−1(σ ) b˙ X b˙ ⊗ A ⊗ A a a −1 b˙ 1 a′ 1 b′ b˙′ = [2] (σA)aρ 2 (g) a ρ 2 (h)bXa′ −1 b˙ 1 a′ 1 b′ b˙′ = [2] (σA)a ρ 2 (g) a ρ 2 (h) b(σµ)a′ Xµ = X Λ(g h)µ , (3.27) µ ⊗ A for any (g h) q(sl2(C)). From this we can read oﬀ explicit formulas for Λ in ⊗ 1∈ U terms of the D 2 -representation of (su ) and the q-Pauli matrices Uq 2 µ −2 −1 b˙ 1 a′ 1 b′ b˙′ Λ(g h) 0 = q [2] (σ0)a ρ 2 (g) a ρ 2 (h) b(σµ)a′ ⊗ (3.28) µ −1 b˙ 1 a′ 1 b′ b˙′ Λ(g h) = [2] (σ ) ρ 2 (g) ρ 2 (h) (σ ) ′ . ⊗ A A a a b µ a The matrices representing the generators of rotations and boosts have been calculated explicitly in Eqs. (A.50) and (A.51).

3.1.3 Commutation Relations of the q-Poincar´eAlgebra In order to construct the q-Poincar´ealgebra we have to view as the algebra Mq of translations, so we write Pµ instead of Xµ. By construction q is a left (sl (C))-module -algebra. Denoting the action of h (sl (C))M on p Uq 2 ∗ ∈ Uq 2 ∈ Mq by h⊲p this means

′ ′ ∗ ∗ ∗ h⊲pp =(h(1) ⊲p)(h(2) ⊲p ) , (h⊲p) =(Sh) ⊲p . (3.29)

As in the undeformed case, q(sl2(C)) and q can then be joined together in a semidirect product: U M

Definition 7. The -algebra of the Hopf semidirect product q ⋊ q(sl2(C)), that is, the vector space∗ (sl (C)) with multiplication M U Mq ⊗ Uq 2 (p h)(p′ h′) := p(h ⊲p′) h h′ (3.30) ⊗ ⊗ (1) ⊗ (2) and -structure (p h)∗ = (1 h∗)(p∗ 1), is , the q-Poincaréalgebra. ∗ ⊗ ⊗ ⊗ Pq 3.1 The q-PoincaréAlgebra 39

By construction we have

µ′ (adLh) ⊲Pµ = h(1)PµS(h(2))= h⊲Pµ = Pµ′ Λ(h) µ (3.31) for all h (sl (C)), that is, P a 4-vector operator. ∈ Uq 2 µ We want to calculate the commutation relations between q-Lorentz generators and momenta explicitly. By construction of the 4-vectors the zero component P0 commutes with all rotations. According to Eq. (2.25) we get for the 3-vector part

EP = P E + q(A+1) [A + 2][1 A] P K A A − A+1 −2A −A FPA = q PAF + q p [1 + A][2 A] PA−1 (3.32) − 2A KPA = q PAK, p where A runs through 1, 0, 1 = , 3, + . In terms of the vectorial generators {− } {− } this becomes

C DE C JAPB = PAJB εA Bε C PDJE + εA B PC W − (3.33) WP =(λ2 + 1)P W λ2εBC P J . A A − A B C

For the commutation relations between momenta and boosts we use Eq. (A.48) to write in an obvious matrix notation

[4] 1 1 −1 −1 − 2 − 2 a b [2] [2] P0 + q λP3 q λ[2] P+ a b P0 = 1 − 1 −1 [4] c d  q 2 λ[2] 2 P [2] P qλP  c d − − [2] 0 − 3  − 1 − 1  a b P− q 2 λ[2] 2 (P3 P0) a b P− = − c d 0 P c d − (3.34) a b P+ 0 a b P+ = 1 − 1 c d q 2 λ[2] 2 (P P ) P c d − 3 − 0 + −1 − 1 − 1 a b [2] (2P3 + qλP0) q 2 λ[2] 2 P+ a b P3 = 1 − 1 −1 −1 . c d q 2 λ[2] 2 P [2] (2P q λP ) c d − − 3 − 0

The commutation relations between momenta and the vectorial RS-generators as deﬁned in Eq. (2.63) are more complicated but involve only 3-vectors and scalars 40 3. Algebraic Structure of the q-Poincar´eAlgebra with respect to rotations. R P = [4][2]−2P R + λ[2]−1εAB P R q[2]−1P U (3.35a) C 0 0 C C A B − C −2 −1 AB −1 −1 SC P0 = [4][2] P0SC + λ[2] ε C PASB + q [2] PC U (3.35b)

R P = qP R λ[2]−1ε B P R q−1λ[2]−1g (gABP R ) C D C D − C D 0 B − CD A B 2[2]−1ε X εAB P R q−1[2]−1g P U + [2]−1ε A P U (3.35c) − C D X A B − CD 0 C D A S P = qP S λ[2]−1ε B P S + qλ[2]−1g (gABP S ) C D C D − C D 0 B CD A B 2[2]−1ε X εAB P S + q[2]−1g P V + [2]−1ε A P V (3.35d) − C D X A B CD 0 C D A

UP = [4][2]−2P U q−1λ2[2]−1(gABP R ) (3.35e) 0 0 − A B −2 0 2 −1 AB VP0 = [4][2] P V + qλ [2] (g PASB) (3.35f)

UP = [4][2]−2P U qλ2[2]−1P R λ2[2]−1 εAB P R (3.35g) C A − 0 A − C A B VP = [4][2]−2P V + q−1λ2[2]−1P S λ2[2]−1 εAB P S (3.35h) C A 0 A − C A B Finally, we want to indicate how one can boost 4-vector operators. Let V0 be some element of the q-Poincar´ealgebra. If we assume that V0 is the zero component of a left 4-vector operator the action of the boosts on V0 must be the same as on P0, so according to Eq. (A.48) we must deﬁne the other components by

− 1 −1 1 V := ad ( q 2 λ [2] 2 c) ⊲V − L − 0 1 −1 1 V+ := adL(q 2 λ [2] 2 b) ⊲V0 (3.36) V := ad (λ−1 (d a)) ⊲V . 3 L − 0 We will make use of this method of computing 4-vectors in Sec. 5.2.2 in order to compute the γ-matrices. In case we know the zero component V0˜ of a right 4-vector the other components must be deﬁned by

1 −1 1 V ˜ := V˜ ⊳ ad ( q 2 λ [2] 2 b) − 0 R − − 1 −1 1 2 2 (3.37) V+˜ := V0˜ ⊳ adR(q λ [2] c) −1 V˜ := V˜ ⊳ ad (λ (d a)) . 3 0 R − 3.2 The q-Pauli-Lubanski Vector and the Spin Casimir 3.2.1 The q-Euclidean Algebra Rotations and translations generate a -subalgebra of the q-Poincar´ealgebra, the ∗ q-Euclidean subalgebra q. Since rotations form a q(su2) Hopf subalgebra of (sl (C)) this q-EuclideanE subalgebra is a semidirectU product Uq 2 = ⋊ (su ) . (3.38) Eq Mq Uq 2 3.2 The q-Pauli-Lubanski Vector and the Spin Casimir 41

By comparing Eq. (3.13) with Eq. (2.29) we note that q and q(su2) are very similar as algebras. One could identify the generators byM a map ξ U: (su ) Mq → Uq 2 with ξ(PA) = αJA, ξ(P0) = βW , for some numbers α, β. More precisely, ξ is a homomorphism of algebras as long as α/β = λ. − We cannot invert ξ, though, since there is no relation like

W 2 λ2J J gAB = 1 (3.39) − A B in . However, for the case of constant positive mass, P P µ = m2, we ﬁnd Mq µ ξ(P P µ)= β2W 2 α2J J gAB = m2 . (3.40) µ − A B We conclude that the image of the constant mass relation in holds in (su ) Mq Uq 2 if α = mλ and β = m. This is consistent with the requirement α/β = λ. We conclude− that / P P µ = m2 is isomorphic to the vectorial form of − (su ). Mq h µ i Uq 2 Setting aside the lack of K−1 in the vectorial (su ) we thus have an isomorphism Uq 2 φ / P P µ = m2 (su ) ⋊ (su ) , (3.41) Eq h µ i −→ Uq 2 Uq 2 where the action of the semidirect product on the right hand side is the left Hopf adjoint action of (su ) on itself. The isomorphism is given by ξ ⋊ 1 on the Uq 2 momenta and 1 ⋊ id on the rotations,

φ(P )= mλJ ⋊ 1 , φ(P )= mW ⋊ 1 (3.42a) A − A 0 φ(JA)=1 ⋊ JA , φ(W )=1 ⋊ W. (3.42b)

Introducing

J := λ−1W, (3.43) 0 − we can write φ more compactly as

φ(P )= mλJ ⋊ 1 , φ(J )=1 ⋊ J , (3.44) µ − µ µ µ where µ runs through 0, , +, 3 . Note, however, that J is no 4-vector operator. { − } µ The introduction of J0 merely allows for a more compact notation. For example, 2 µ Eq. (3.39) can be written as λ JµJ = 1. Furthermore, it is convenient to give the pre-image of K a name

π := mφ−1(K)=(P P ) . (3.45) 0 − 3 For the semidirect product of a Hopf algebra H with itself by the left Hopf adjoint action we have the following isomorphism of algebras

ψ : H ⋊ H H H , ψ(g ⋊ h)= gh h . (3.46) −→ ⊗ (1) ⊗ (2) 42 3. Algebraic Structure of the q-Poincar´eAlgebra

First, we prove that ψ is a homomorphism ψ[(g ⋊ h)(g′ ⋊ h′)] = ψ[g(h ⊲g′) ⋊ h h′]= gh g′S(h )h h′ h h′ (1) (2) (1) (2) (3) (1) ⊗ (4) (2) = gh g′h′ h h′ =(gh h )(g′h′ h′ ) (1) (1) ⊗ (2) (2) (1) ⊗ (2) (1) ⊗ (2) = ψ(g ⋊ h) ψ(g′ ⋊ h′) . (3.47) The invertibility can be shown directly, by deﬁning ψ−1(g h) := gS(h ) ⋊ h , (3.48) ⊗ (1) (2) and checking that (ψ ψ−1)(g h)= ψ[gS(h ) ⋊ h ]= gS(h )h h = g h ◦ ⊗ (1) (2) (1) (2) ⊗ (3) ⊗ (3.49) (ψ−1 ψ)(g ⋊ h)= ψ[gh h ]= gh S(h ) ⋊ h = g ⋊ h . ◦ (1) ⊗ (2) (1) (2) (3) Thus, Eq. (3.46) tells us, that we have the sequence of Isomorphisms

φ ψ / P P µ = m2 (su ) ⋊ (su ) (su ) (su ) . (3.50) Eq h µ i −→ Uq 2 Uq 2 −→ Uq 2 ⊗ Uq 2 Through these isomorphisms we get a full understanding of the structure of the q-Euclidean algebra. One particularly interesting fact is that there is a whole q(su2) subalgebra of which commutes with the momenta . This subalgebraU is embedded by Eq Mq the map i : (su ) , i = φ−1 ψ−1 (1 id) , (3.51) Uq 2 −→ Eq ◦ ◦ ⊗ which computes to −1 −1 i(J±)= J± + λ P±π K i(J )= mλ−1π−1K m−1(λ−1P W + gABP J ) 3 − 0 A B −1 AB (3.52) i(W )= m (P0W + λg PAJB) i(K)= m π−1K.

Observe that the images of JA do not exist in q proper, since they all involve the inverse of π = P P , which is not an elementE of . 0 − 3 Mq 3.2.2 The Center of the q-Euclidean Algebra

µ 2 We wonder where precisely the condition PµP = m has entered into our considerations. Which of the results do still hold if the mass shell condition is relaxed? Towards this end we list the commutation relations between rotations and translations

−1 [J−,P−] = 0 [J−,P3]= q P−K [J−,P+]= P3K [J ,P ]= P K [J ,P ]= qP K [J ,P ] = 0 (3.53) + − − 3 + 3 − + + + [K,P ]= q−1λP K [K,P ] = 0 [K,P ]= qλP K − − − 3 + + 3.2 The q-Pauli-Lubanski Vector and the Spin Casimir 43

Let us check what relations still hold within i( (su )). We compute for example Uq 2 −1 −1 −1 i(K)i(J+)= mπ K(J+ + λ P+π K) 2 −1 −1 −1 −1 2 =(q J+ + qP+π K)mπ K + mλ P+(π K) 2 −1 −1 −1 =(q J+ +(q + λ )P+π K)mπ K 2 = q i(J+)i(K) , (3.54)

2 telling us that the relation KJ+ = q J+K is preserved under i. Similarly, we ﬁnd that the image of KJ = q−2J K still holds in i( (su )). Hence, we did not − − Uq 2 use the mass shell condition for these two relations. However, for the relation λ[2](qJ J q−1J J )=1 K2 we ﬁnd + − − − + − P P µ i[λ[2](qJ J q−1J J )]=1 µ i(K)2 , (3.55) + − − − + − m2 µ such that this relation holds in q precisely if the mass shell condition PµP = m2 holds. We conclude that withoutE the mass shell condition i is no longer a homomorphism of algebras. Now, we check if i( q(su2)) still commutes with all translations. Setting aside the problem that π−1 doesU not exist in proper, we compute for example Eq −1 −1 −1 −1 P+i(J−)= P+(J− + λ P−π K)= J−P+ +(λ P+P− P3π)π K −1 −1 − = J−P+ + λ (P−P+)π K = i(J−)P+ . (3.56)

In the same manner we ﬁnd, that all of i( q(su2)) commutes with all translations. This holds in particular for i(W ) whichU is furthermore a scalar with respect to rotations, since it is made up of the scalars P0, W , and P~ J~. In conclusion we have3 · Proposition 3. The center of the q-Euclidean algebra is generated by P P µ, Eq µ P0 and Z := m i(W )= P W + λgABP J = λP J µ . (3.58) 0 A B − µ 3.2.3 The Pauli-Lubanski Vector in the q-Deformed Setting In the undeformed case one considers the Pauli-Lubanski (pseudo) vector 1 W q=1 := ε V νσP τ , (3.59) µ −2 µνστ where V νσ is the matrix of Lorentz generators. Its usefulness is due to the following two properties: 3 Using the Casimir operators of q, the orbital angular momentum relation of [31] can be equivalently written as E

λ(P~ J~)= P0(1 W ) Z = P0 . (3.57) · − ⇔ 44 3. Algebraic Structure of the q-Poincar´eAlgebra

(i) Wµ is a 4-vector operator of the Poincar´ealgebra.

τ (ii) Each component Wµ commutes with all translations P .

If we demand further that Wµ be linear in the Lorentz generators and the translations, conditions (i) and (ii) determine the Pauli-Lubanski vector up to a constant µ factor. From (i) and (ii) we deduce that WµW is a Casimir operator. Physically, this Casimir operator turns out to correspond to spin. In the q-deformed case we are tempted to deﬁne Wµ analogously by Eq. (3.59) with the q-deformed versions of the epsilon tensor, the matrix of Lorentz generators, and the translations. By construction, this would be a 4-vector operator. However, it turns out that with this naive approach property (ii) will not hold. Therefore, we will try to ﬁnd a way to construct Wµ such that (ii) holds, as well. Let us start with the zero component W0. It has to commute with all translations to satisfy (ii) and with all rotations since the zero component of a 4-vector is a scalar with respect to rotations. Thus, it has to commute with all of the q-Euclidean algebra . If we assume that as in the undeformed case W is it- Eq 0 self a member of q, we conclude that W0 has to be an element of the center of the q-Euclidean algebra,E which we computed in the preceding section. Since the momenta carry dimensions W0 has to be linear in the momenta. Hence W0 must be a linear combination of P0 and Z. The additional requirement that W0 has to have the right undeformed limit determines

W := λ−1(Z P )= λ−1(W 1)P + gABJ P (3.60) 0 − 0 − 0 A B up to an overall factor that tends to one as q 1. → Now that we have a good candidate for the zero component of the q-Pauli- Lubanski vector we have to see if it can be boosted to a 4-vector. First we have to ask what type of vector operator we would expect it to be. Recall from Sec. 2.2.1 that we have to distinguish between left and right tensor operators. A short calculation shows that for any translation p q and any Lorentz transformation h (sl (C)) ∈ M ∈ Uq 2

(W0 ⊳ adRh) p = S(h(1))W0h(2)p = S(h(1))W0(h(2) ⊲p)h(3)

= S(h(1))(h(2) ⊲p)W0h(3) =(S(h(1))(1)h(2) ⊲p)S(h(1))(2)W0h(3)

=(S(h(2))h(3) ⊲p)S(h(1))W0h(4)

= p (W0 ⊳ adRh) (3.61)

Hence, a right boosted W0 commutes with all translations. This is not be the case for adL ⊲ W0. Hence, the q-Pauli-Lubanski vector will satisfy property (ii) only if it is a right vector operator Wµ˜. 3.2 The q-Pauli-Lubanski Vector and the Spin Casimir 45

3.2.4 Boosting the q-Pauli-Lubanski Vector

If W0 = W0˜ as deﬁned in (3.60) really is a left 4-vector operator, which is not necessarily so, then the other components are given, uniquely, by Eqs. (3.37). We will now determine Wµ˜ and rigorously show that it is a right 4-vector operator. Up to a constant factors W0˜ is the sum of two parts, Z and P0, which we will treat separately.

Boosting Z The explicit calculations of the right adjoint action of the boosts on Z by Eqs. (3.37) turn out to be very lengthy. It is more efficient to start with a more abstract consideration. We observe that for all boosts h SU (2)op we have ∈ q µ′ (J ) , h (J ) = J ′ Λ(h) , (3.62) h µ (1) i µ (2) µ µ where , is the dual pairing of (su ) and SU (2). We exemplify this result h· ·i Uq 2 q for J+, (J ) , Ba (J ) = J , Ba K + 1, Ba J h + (1) bi + (2) h + bi h bi + = λ[2]−1(σ ) (J J )+ δa J + ab 3 − 0 b + J+ 0 = 1 −1/2 q 2 λ[2] (J J ) J − 3 − 0 + µ′ = Jµ′ Λ(h) + . (3.63) Applying the map φ−1 as defined in Eq. (3.44) to Eq. (3.62) we get ad h⊲φ−1(l)= l , h φ−1(l ) . (3.64) L h (1) i (2) for all l (su ) and h SU (2)op. For example, for l = φ(P ) = mλJ the ∈ Uq 2 ∈ q + − + left adjoint action of the boost generators on P+ can be written as ad Ba ⊲P = δa P λ J , Ba π = δa P + λ[2]−1 (σ ) (P P ) , (3.65) L b + b + − h + bi b + + ab 3 − 0 which is the same as in Eq. (A.48). Let i be the map that has been defined in Eq. (3.51). We try to commute i(l), l (su ), with a boost h SU (2)op using Eq. (3.64): ∈ Uq 2 ∈ q i(l) h = φ−1[S(l )]l h = φ−1[S(l )]h l S(l ), h l , h (1) (2) (1) (2) (3)h (2) (1)ih (4) (3)i = h ad S−1(h ) ⊲φ−1[S(l )] l S(l ), h l , h (3){ L (2) (1) } (3)h (2) (1)ih (4) (4)i = h S(l ),S−1(h ) φ−1[S(l )]l S(l ), h l , h (3)h (2) (2) i (1) (4)h (3) (1)ih (5) (4)i = h φ−1[S(l )]l l , h (3.66) (1) (1) (2)h (3) (2)i This leads to a remarkably simple formula for the right adjoint action of a boost on i(l) i(l) ⊳ ad h = i(l l , h ) . (3.67) R (1)h (2) i 46 3. Algebraic Structure of the q-PoincaréAlgebra

For l = S(Jµ) this formula becomes i[S(J )] ⊳ ad h = i[S(J ) S(J ) , h ]= i[S((J ) (J ) ,S−1h )] µ R µ (1)h µ (2) i µ (2)h µ (1) i −1 µ′ = i[S(Jµ′ )Λ(S h) µ] , (3.68) which tells us that i(S(Jµ)) transforms under boosts as a right lower 4-vector operator. It remains to check whether i(S(Jµ)) transforms as right 4-vector under ro- −1 tations. We observe that φ maps the 3-vector Jµ to the 3-vector Pµ and the (su )-scalar J to the scalar P . Hence, for a, b (su ) we have Uq 2 0 0 ∈ Uq 2 −1 −1 adLb⊲φ (a)= φ (adLb ⊲ a) . (3.69) Now we are prepared to tackle the right action of a rotation on i(Sa) −1 i(Sa) ⊳ adRb = S(b(1))φ [S((Sa)(1))](Sa)(2)b(2) −1 = S(b(1))(1) ⊲φ [S((Sa)(1))] S(b(1))(2)(Sa)(2)b(2) {−1 }{ } = φ [(S(b(2))(1)S((Sa)(1))S(S(b(2))(2))]S(b(1))(Sa)(2)b(3) −1 = φ [S(S(b(2))(Sa)(1)b(3))]S(b(1))(Sa)(2)b(4) −1 = φ [S(S(b(1))(1)(Sa)(1)b(2)(1))]S(b(1))(2)(Sa)(2)b(2)(2) −1 = i(Sa⊳ adRb)= i(S(adLS b ⊲ a)) . (3.70)

This shows that since S(Jµ) transforms as a right lower 4-vector under rotations, so does i(S(Jµ)). In conclusion we have Proposition 4. The set of operators Z := mλ i(S(J )) (3.71) µ˜ − µ is a right lower 4-vector operator of the q-Lorentz algebra. Since furthermore

Z0˜ = Z, Zµ˜ is the unique right lower 4-vector operator with zero component Z.

All that remains to do is to compute Zµ˜ explicitly: AB Z0˜ = m i(W )= P0W + λg PAJB −1 Z±˜ = P± + λJ±K π (3.72) −1 AB −1 Z˜ = m i(W K )= P W + λg P J K π . 3 − 0 A B − Observe that these expressions do not contain π−1, hence, they are proper mem- bers of , that is, Eq −1 Z0˜ = WP0 qλJ+P− q λJ−P+ + λJ3P3 − −1 − Z±˜ = P± + λJ±K (P0 P3) (3.73) −1 − −1 −1 Z˜ =(W K )P qλJ P q λJ P +(λJ + K )P . 3 − 0 − + − − − + 3 3 Finally, we recall that the square of Zµ˜ must be a Casimir operator. After lengthy calculations we ﬁnd µ˜ µ Z Zµ˜ = PµP . (3.74)

We conclude that squaring Zµ˜ alone does not yield a new Casimir operator. 3.2 The q-Pauli-Lubanski Vector and the Spin Casimir 47

Boosting P0 The next step in our calculation of the q-Pauli-Lubanski vector is to ﬁnd a right 4-vector operator with P as zero component. With a universal - 0 R matrix of the q-Lorentz algebra, we can generically turn a left 4-vector operator into a right 4-vector operator. Deﬁning for the left 4-vector operator Pµ

2 2 µ′ j(P ) := S ( )( ⊲P )= S Λ( ) P ′ (3.75) µ R[1] R[2] µ R[1] R[2] µ µ we check that for any q-Lorentz transformationh (sl (C)) we have ∈ Uq 2 j(P ) ⊳ ad h = S(h )S2( )( ⊲P )h µ R (1) R[1] R[2] µ (2) = S(h )S2( )h S−1(h ) ⊲P (1) R[1] (3) (2) R[2] µ = S S( )h h S−1(S( )h ) ⊲P R[1] (1) (3) R[2] (2) µ = S h S( ) h S−1(h S( )) ⊲P (2) R[1] (3) (1) R[2] µ = S2( )S(h )h S−1(h ) ⊲P R[1] (2) (3) R[2] (1) µ = S2( )( S−1h⊲P ) R[1] R[2] µ −1 µ′ = j(Pµ′ )Λ(S h) µ , (3.76) thus, j(Pµ) is indeed a right 4-vector operator. Recall from Sec. 2.3.2, that the object

(LΛ )µ := Λ( )µ (3.77) + ν R[1] R[2] ν that appears in the deﬁnition of j(Pµ) is an L-matrix. Furthermore, we recall from Eq. (1.53) that there are two universal -matrices of the q-Lorentz algebra, which are composed of the -matrix of (slR) according to R Uq 2 = −1 −1 , = −1 . (3.78) RI R41 R31 R24R23 RII R41 R13R24R23 1 1 We will now compute the L-Matrix for I. We have for the ( 2 , 2 )-form of the vector representation R

1 1 ( , ) ab 1 1 ab L 2 2 = id id ρ 2 ρ 2 ( ) I+ cd ⊗ ⊗ ⊗ RI cd 1 b 1 a 1 b′ 1 a′ = L 2 ′ L 2 ′ L 2 L 2 − b − a ⊗ + d + c b a = B dB c, (3.79) where Ba SU (2)op is the matrix of boosts. For the 4-vector form of this b ∈ q L-matrix we then ﬁnd 10 0 0 2 2 1 1 Λ µ 0 a b q 2 [2] 2 ab (LI+) ν =  2 2 1 1  (3.80) 0 c d q 2 [2] 2 cd 1 1 1 1  2 2 2 2  0 q [2] ac q [2] bd (1 + [2]bc)   48 3. Algebraic Structure of the q-Poincar´eAlgebra with respect to the basis 0, , +, 3 . This matrix of generators becomes more familiar if we write it in block{ − diagonal} form

1 0 (LΛ )µ = , (3.81) I+ ν 0 tA B A so we can see that t B, A, B 1, 0, 1 is the 3-dimensional corepresentation op ∈ {− } matrix of SUq(2) [52,53]. From the block diagonal form we deduce that

j(P0)= P0 , (3.82) so we get

Proposition 5. The set of operators

′ 2 Λ µ ′ j(Pµ) := S (LI+) µ Pµ (3.83) is a right lower 4-vector operator of the q-Lorentz algebra. Since furthermore j(P0)= P0, j(Pµ) is the unique right lower 4-vector operator with zero component P0.

With Eq. (3.83) we ﬁnd the explicit expressions

j(P0)= P0 2 −4 2 − 3 1 j(P−)= a P− + q c P+ + q 2 [2] 2 ac P3 (3.84) 4 2 2 5 1 j(P+)= q b P− + d P+ + q 2 [2] 2 bd P3 5 1 − 3 1 j(P3)= q 2 [2] 2 ab P− + q 2 [2] 2 cdP+ + (1 + [2]bc) P3 .

Finally, we want to calculate the square of j(Pµ) which must be a Casimir operator. First, we note that since P0 commutes with all momenta and j(Pµ) is the right boosted P0, the reasoning of Eq. (3.61) applies, that is, all momenta Pµ commute with j(Pν),

Pµ j(Pν)= j(Pν) Pµ . (3.85)

Moreover, we have

Λ µ Λ σ τν ′ µ ′ σ τν (LI+) ν(LI+) τ η = R[1]R[1 ]Λ(R[2]) νΛ(R[2 ]) τ η µ σ′′ τν σσ′ = R[1]R[1′]Λ(R[2]) νΛ(R[2′]) τ η ησ′σ′′ η µ −1 ν σσ′ = R[1]R[1′]Λ(R[2]) νΛ(S R[2′]) σ′ η ′ −1 −1 µ ′ σσ = R[1]R[1′]Λ(R[2]R[2′]) σ η = ησµ , (3.86) 3.3 The Little Algebras 49 where we have used Eq. (2.13). With the last two equations we can compute the square of j(Pµ) quite easily

′ ′ µ 2 Λ µ ′ 2 Λ ν ′ νµ j(P )j(Pµ)= S (LI+) µ Pµ S (LI+) ν Pν η ′ ′ 2 Λ µ 2 Λ ν νµ ′ ′ = S (LI+) µ S (LI+) ν η Pν Pµ νµ = η PνPµ . (3.87) Again, the square of one half of the q-Pauli-Lubanski vector alone yields only the mass Casimir.

The q-Pauli-Lubanski Vector We come to the following conclusion: Proposition 6. The set of operators W := λ−1[Z j(P )] = m i(S(J )) λ−1j(P ) (3.88) µ˜ µ˜ − µ − µ − µ has the following properties: (i) It is a right lower 4-vector operator.

(ii) Each component Wµ commutes with all translations Pτ . Furthermore, it is the unique right lower 4-vector operator with zero component W = λ−1(Z P ). We will therefore call it the q-Pauli-Lubanski vector. 0 − 0 Explicitly, the q-Pauli-Lubanski vector is

−1 −1 W˜ = λ (W 1)P qJ P q J P + J P 0 − 0 − + − − − + 3 3 −1 −1 2 −4 2 −1 − 3 1 W ˜ = λ [λJ K P + (1 a )P q c P (λJ K + q 2 [2] 2 ac)P ] − − 0 − − − + − − 3 −1 −1 4 2 2 −1 5 1 W ˜ = λ [λJ K P q b P + (1 d )P (λJ K + q 2 [2] 2 bd)P ] + + 0 − − − + − + 3 −1 −1 5 −1 1 −1 − 3 −1 1 W˜ = λ (W K )P (qJ + q 2 λ [2] 2 ab)P (q J + q 2 λ [2] 2 cd)P 3 − 0 − + − − − + +(J + λ−1K−1 λ−1(1 + [2]bc))P . 3 − 3 (3.89)

3.3 The Little Algebras 3.3.1 Little Algebras in the q-Deformed Setting In classical relativistic mechanics the state of a free particle is completely determined by its 4-momentum. In quantum mechanics particles can have an additional degree of freedom called spin. What is spin? Let us assume we have a free relativistic particle described by an irreducible representation of the Poincar´ealgebra. We pick all states with a given momentum, := ψ : P ψ = p ψ , (3.90) Hp {| i ∈ H µ| i µ| i} 50 3. Algebraic Structure of the q-Poincar´eAlgebra

where is the Hilbert space of the particle and p = (pµ) is the 4-vector of momentumH eigenvalues. If the state of the particle is not uniquely determined by the eigenvalues of the momentum, then the eigenspace p will be degenerate. In that case we need, besides the momentum eigenvalues,H an additional quantity to label the basis of our Hilbert space uniquely. This additional degree of freedom is spin. The spin symmetry is then the set of Lorentz transformations that leaves the momentum eigenvalues invariant and, hence, acts on the spin degrees of freedom only,

′ := h : P h ψ = p h ψ for all ψ , (3.91) Kp { ∈L µ | i µ | i | i ∈ Hp} ′ where is the enveloping Lorentz algebra. In mathematical terms, p is the L ′ K stabilizer of p. Clearly, p is an algebra, called the little algebra. A priori,H there are aK lot of different little algebras for each representation and each vector p of momentum eigenvalues. In the undeformed case it turns out that for the physically relevant representations (real mass) there are (up to isomorphism) only two little algebras, depending on the mass being either positive or zero [1]. For positive mass we get the algebra of rotations, (su2), for zero mass an algebra that is isomorphic to the algebra of rotations andU translations of the 2- ′ dimensional plane denoted by (iso2). The proof that p does not depend on the particular representation but onU the mass, does not generalizeK to the q-deformed case: If we defined for representations of the q-Poincaréalgebra the little algebra ′ 1 as in Eq. (3.91), it could well happen that p for a spin- 2 particle is not the same as for spin-1. We will therefore define theKq-little algebras differently. In the undeformed case there is an alternative but equivalent definition of the ′ little algebras. p is the algebra generated by the components of the q-Pauli- Lubanski vectorK as defined in Eq. (3.59) with the momentum generators replaced by their eigenvalues. Let us formalize this to see why this definition works and how it is generalized to the q-deformed case. Let be the algebra of translations, the Lorentz algebra, both joined in a semidirectT product to form the PoincaréalgebraL = ⋊ . Let χ be P T L p the map that maps the momentum generators to the eigenvalues, χp(Pµ) = pµ. Being the restriction of a representation, χp must extend to a one dimensional -representation χp : C, a non-trivial condition only in the q-deformed case.∗ Noting that everyT element→ of can be written as a sum of products of P Lorentz transformations and translations, i liti, we extend χp to a linear map χ˜p : by P→L P

χ˜p( liti) := liχp(ti). (3.92)

The little algebra can now be alternativelyX X deﬁned as the unital algebra generated by the images of the q-Pauli-Lubanski vector underχ ˜p, := C χ˜ (W ) . (3.93) Kp h p µ i 3.3 The Little Algebras 51

Why is this a reasonable deﬁnition? By construction the action of every element of on is the same as of its image underχ ˜ . For any ψ this means P Hp p | i ∈ Hp P χ˜ (W ) ψ =χ ˜ (P W ) ψ =χ ˜ (W P ) ψ = p χ˜ (W ) ψ , (3.94) µ p ν | i p µ ν | i p ν µ | i µ p ν | i ′ which shows that p p. It still could happen, that p is strictly smaller ′ K ⊂ K K than p. In the undeformed case there are theorems telling us [54, 55] that this K ′ cannot happen, so we really have p = p. For the q-deformed case no such theorem is known [56]. However, ifK thereK were more generators in the stabilizer of some momentum eigenspace they would have to vanish for q 1. In this sense Eq. (3.93) with the q-deformed Pauli-Lubanski vector can be considered→ to deﬁne the q-deformed little algebras.

3.3.2 Computation of the q-Little Algebras To begin the explicit calculation of the q-deformed little algebras, we need to figure out if there are eigenstates of q-momentum at all. That is, we want to determine the one-dimensional -representations of , that is the homomor- ∗ Mq phisms of -Algebras χ : C. Let us again denote the eigenvalues of the ∗ Mq 7→ generators by lower case letters pµ := χ(Pµ). For χ to be a -map we must have ∗ ∗ p0, p3 real and p+ = qp−. To find the conditions for χ to be a homomorphism of algebras, we apply−χ to the relations (3.12) of , yielding Mq p (p p )=0 . (3.95) A 0 − 3 There are two cases. The first is p0 = p3, which immediately leads to pA = 0, and p = m. The second case is p = 6p , leading to m2 = p 2 p 2. Hence, if 0 ± 0 3 −| −| −| +| the mass m is to be real, we must have p± = 0. To summarize, for real mass m we have a massive and a massless type of momentum eigenstates with eigenvalues given by

( m, 0, 0, 0) m> 0 (p0,p−,p+,p3)= ± (3.96) (k, 0, 0,k) m =0, k R ( ∈ Now, we need to move the momentum generators in the expressions of the q- Pauli-Lubanski vector to the right and replace them with these eigenvalues.

The Massive Case In Eqs. (3.89) the momenta have already been moved to the right, so we can simply replace them with (P0,P−,P+,P3) (m, 0, 0, 0). We get → −1 χ˜p(W0˜)= λ (W 1)m −1 − χ˜p(W−˜ )= J−K m −1 (3.97) χ˜p(W+˜ )= J+K m −1 −1 χ˜ (W˜)= λ (W K )m , p 3 − 52 3. Algebraic Structure of the q-Poincar´eAlgebra

−1 −1 so the set of generators of the little algebra is essentially W, K ,J±K . Since K−1 stabilizes the momentum eigenspace, so does its inverse{ K. Hence, it} is safe to add K to the little algebra which would exist, anyway, as operator within a representation. We thus get := = (su ) , (3.98) Km K(m,0,0,0) Uq 2 completely analogous to the undeformed case.

The Massless Case The massless case is more complicated. Replacing in Eqs. (3.89) the momentum generators with (P ,P ,P ,P ) (k, 0, 0,k) we get 0 − + 3 → −1 χ˜ (W˜)= λ (K 1)k p 0 − −1 − 3 1 χ˜p(W ˜ )= λ q 2 [2] 2 ac k − − (3.99) −1 5 1 χ˜ (W ˜ )= λ q 2 [2] 2 bd k p + − −1 χ˜ (W˜)= λ K (1 + [2]bc) k . p 3 − The set of generators of the little algebra is essentially K,ac,bd,bc . The commutation relations of these generators can be written more{ conveniently} in terms of K and 1 1 1 1 N− := q 2 [2] 2 ac , N+ := q 2 [2] 2 bd , N3 := 1 + [2]bc , (3.100) 3 A or equivalently NA = t A, for t B as defined by Eqs. (3.80) and (3.81). The commutation relations are N N εAB = λN , N N gBA =1 , KN = q−2AN K, (3.101) B A C − C A B A A and the conjugation properties ∗ BA ∗ NA = NB g , K = K. (3.102) op In words: The NA generate the opposite algebra of a unit quantum sphere, q∞ [57]. K, the generator of (u ), acts on N as on a right 3-vector operator.S In Uq 1 A total we have := = (u ) ⋉ op . (3.103) K0 K(k,0,0,k) Uq 1 Sq∞ As opposed to the massive case, is no Hopf algebra. However, since N = t3 K0 A A and ∆(tA )= tA tB , we have C B ⊗ C ∆(N )= N tA , (3.104) B A ⊗ B hence, is a right coideal. K0 The only irreducible -representations of 0 are one-dimensional. They depend on a real parameter∗α and are defined onK the single basis vector α by | i K α = α α , N α =0 , N α = α . (3.105) | i | i ±| i 3| i | i Unlike for the undeformed case, no infinite-dimensional irreducible representation exists. Chapter 4 Massive Spin Representations

4.1 Representations in an Angular Momentum Basis 4.1.1 The Complete Set of Commuting Observables We want to construct a massive irreducible representations in a basis that can be given a physical interpretation. Massive irreducible means that within the representation we have

µ 2 PµP = m (4.1)

µ for some real positive constant m, PµP being the mass Casimir operator. We have shown in Sec. 3.3.2 that there are rest states, that is, momentum eigenstates, Pµ ψ0 = pµ ψ0 , with (pµ)=(p0,p−,p+,p3)=(m, 0, 0, 0). On these rest states the| q-Pauli-Lubanskii | i vector acts as

−1 W˜ ψ = mλ (W 1) ψ , W ˜ ψ = mS(J ) ψ , (4.2) 0| 0i − | 0i A| 0i − A | 0i from which it follows that

W µ˜W ψ =2m2λ−2(1 W ) ψ . (4.3) µ˜| 0i − | 0i µ˜ The spin Casimir W Wµ˜ must be constant, thus, the angular momentum must be constant within the rest frame. According to Eq. (A.33) the possible values are

W ψ = [2]−1 q(2s+1) + q−(2s+1) ψ , (4.4) | 0i | 0i where s 1 N is a half integer. For the spin Casimir this means ∈ 2 0 W µ˜W ψ = 2[2]−1m2[s + 1][s] ψ . (4.5) µ˜| 0i − | 0i In accordance with the undeformed case we will call s the spin of the representation. The space of all rest states is stabilized by the algebra generated by the little algebra for the massive case, (su ), and the momenta, that is, by the q- Uq 2 Euclidean algebra . The observables that are most commonly diagonalized are Eq all elements of : energy P , momentum P~ , angular momentum J~, helicity J~ P~ . Eq 0 · 54 4. Massive Spin Representations

We opt for an angular momentum basis, where we diagonalize J and J~2 = J~ J~. 3 · If we add the Casimir operators of q, P0 and Z as defined in Eq. (3.58), we E 1 2 get a complete set of commuting observables. Instead of J3 and J~ it is more practical to work with K and W , whose possible eigenvalues can be looked up in Sec. A.2.1. From Sec. 3.2.1 we know that P0 and Z are Casimir operators of a q(su2) algebra, so we know their possible eigenvalues, as well. Labeling the statesU of the yet to be constructed representation by their possible eigenvalues we get including the Casimirs K j, m, n, k = q2m j, m, n, k (4.6a) | i | i W j, m, n, k = [2]−1 q(2j+1) + q−(2j+1) j, m, n, k (4.6b) | i | i P j, m, n, k = m[2]−1 q(2n+1) + q−(2n+1) j, m, n, k (4.6c) 0| i | i Z j, m, n, k = m[2]−1 q(2k+1) + q−(2k+1) j, m, n, k (4.6d) | i | i P P µ j, m, n, k = m2 j, m, n, k (4.6e) µ | i | i W µ˜W j, m, n, k = 2[2]−1m2[s + 1][s] j, m, n, k . (4.6f) µ˜| i − | i The eigenvalues of W , P0, and Z are all of the same form, ξ(j), mξ(n), and mξ(k), where ξ(j) := [2]−1 q(2j+1) + q−(2j+1) . (4.7) For the operators with a more obvious undeformed limit J , J~2, and J~ P~ we get 3 · J j, m, n, k = qm[m] λ[2]−2[2j + 2][2j] j, m, n, k 3| i − | i 2 −2 J~ j, m, n, k = [2] [2j + 2][2j] j, m, n, k | i | i (4.8) (J~ P~ ) j, m, n, k = λ[2]−2 [n + j + k + 2][n + j k] · | i − + [n j + k][n j k] j, m, n, k , − − − | i which shows why it is more efficient to work with K, W , and Z instead. One further advantage of using an angular momentum basis is, that the q- Wigner-Eckart theorem of Page 22 applies. The problem of finding the matrix elements of 3-vector or scalar operators with respect to rotations is reduced to finding the reduced matrix elements. For 3-vector operators such as PA, JA, RA, and SA we get j′, m′, n′,k′ P j, m, n, k = C (1, j, j′ A, m, m′) j′, n′,k′ P~ j, n, k , (4.9) h | A| i q | h k k i while for scalars with respect to rotations such as Z, W , U, and V we get ′ ′ ′ ′ ′ ′ ′ j , m , n ,k Z j, m, n, k = δ ′ δ ′ j , n ,k Z j, n, k . (4.10) h | | i mm jj h k k i The values of the q-Clebsch-Gordan coefficients that we will need are given in Sec. A.1.1. Useful relations for the reduced matrix elements can be derived from Eq. (2.21), which has been done explicitly in Eqs. (A.19). 1The authors of [34,35] failed to add Z or J~ P~ to their set of commuting observables (cf. [35], p. 67). This is the reason why they only found· spin zero representations. 4.1 Representations in an Angular Momentum Basis 55

4.1.2 Representations of the q-Euclidean Algebra If we keep n and k constant, we fix the eigenvalues of the Casimirs operators P and Z of the q-Euclidean algebra . For constant n, k we must thus get an 0 Eq irreducible representation of q. This irreducible representation of q on the mass shell is by isomorphism (3.50)E simply the product Dn Dk of two representationsE ⊗ of (su ). We describe them briefly in terms of reduced matrix elements. Uq 2 The reduced matrix element of JA can be read off Eq. (A.32),

j J~ j = [2]−1 [2j + 2][2j] . (4.11) h k k i − p Due to the Clebsch-Gordan series (2.2) j takes on the values k n , k n + 1,...,k + n . Taking the matrix elements of Eq. (4.6d) we ﬁnd{| − | | − | } [k + n + j + 2][j k + n] [k + n j][j + k n] j, n, k P~ j, n, k = mλ − − − − . h k k i [2] [2j + 2][2j] (4.12) p AB If we take the diagonal matrix elements of the relation PAPB ε C = λP0PC and of Eq. (4.6f) we get, using Eqs. (A.19), two equations for the reduced− matrix elements from which we can eliminate the j P~ j 1 j 1 P~ j term h k k − ih − k k i [2] [2j + 3][2j + 1] j P~ j +1 j +1 P~ j = h k k ih k k i p [2j] j P~ j 2 + λE [2j + 2][2j] j P~ j [2j + 2](P 0P 0 m2) . (4.13) h k k i h k k i − − Upon inserting Eq. (4.12), p

j P~ j +1 j +1 P~ j = h k k ih k k i [k + n + j + 2][k + n j][k n + j + 1][n k + j + 1] m2λ2 − − − , (4.14) − [2][2j + 2] [2j + 3][2j + 1] and using Eq. (A.19e) we ﬁnally get p

j +1,n,k P~ j, n, k = h k k i [k + n + j + 2][k + n j][k n + j + 1][n k + j + 1] mλ − − − (4.15a) p [2][2j + 3][2j + 2] j 1,n,k P~ j, n, k = p h − k k i [k + n + j + 1][k + n j + 1][k n + j][n k + j] mλ − − − . (4.15b) − [2][2j][2j 1] p − p 56 4. Massive Spin Representations

4.1.3 Possible Transitions of Energy and Helicity Next, we will determine the possible transitions of the quantum numbers n and k under the action of the non-Euclidean generators. To ﬁnd restrictions on the possible transitions we consider Eq. (3.35e) and the contraction of Eq. (3.35a) DC with g PD from the left [2]2UP 0 = [4]P 0U q−1λ2[2](P~ R~) (4.16a) − · [2](P~ R~ )P =2P 0(P~ R~) q(P~ P~ )U. (4.16b) · 0 · − · Taking the matrix elements of these equations yields a system of linear equations

0= m [4]ξ(n′) [2]2ξ(n) U q−1λ2[2] P~ R~ (4.17a) − h i − h · i 0= m2q(1 ξ(n′)2) U + m 2ξ(n′) [2]ξ(n) P~ R~ , (4.17b) − h i − h · i where we have used the abbreviation U := j, m, n′,k′ U j, m, n, k and anal- h i h | | i ogously for P~ R~ . For a nontrivial solution to exist, the determinant of the coeﬃcient matrixh · musti vanish,

0 =! m2[2]2(ξ(n′)2 [2]ξ(n′)ξ(n′)+ ξ(n)2)+ m2λ2 − = m2λ4 n + 1 n′ n 1 n′ n + n′ + 1 n + n′ + 3 , (4.18) 2 − − 2 − 2 2 which is, since n 0, precisely the case for n′ = n 1 . ≥ ± 2 To obtain conditions on the transitions of k we contract Eqs. (3.35g) and DC (3.35a) with g JD from the left [2]2U(J~ P~ ) = [4](L~ P~ )U qλ2[2]P (L~ R~) + iλ2[2] J~ (P~ R~) (4.19a) · · − 0 · · × [2](J~ R~ )P = q[2](J~ P~ )U + [4]P (J~ R~) iλ[2] J~ (P~ R~ ) . (4.19b) · 0 − · 0 · − · × Contracting Eq. (3.35c) with J BJ A from the right and eliminating the P~ R~ term · using Eq. (4.16a) yields

λ(J~ R~)(J~ P~ )= q2 qλ[2] (J~ P~ ) λ2WP (J~ P~ )+(J~ J~)UP · · { · − 0} · · 0 q(J~ J~)P λW (J~ P~ ) U 2iq−1[2]−1λW J~ (P~ R~ ) . (4.19c) −{ · 0 − · } − · × Eliminating the J~ (P~ R~ ) term from the last three equations we obtain · × λ2 (J~ R~)Z qZ(J~ R~ ) = q(P WZ)U U(P WZ) (4.20a) { · − · } 0 − − 0 − λ2 (J~ R~)P q−1P (J~ R~ ) = q−1(Z WP 0)U U(Z WP 0) . (4.20b) { · 0 − 0 · } − − − Again we take the matrix elements of these two equations 0= [ξ(n) ξ(j)ξ(k)] q[ξ(n′) ξ(j)ξ(k′)] U +λ2 ξ(k) qξ(k′) J~ R~ { − − − }h i { − }h · i 0= q[ξ(k) ξ(j)ξ(n)] [ξ(k′) ξ(j)ξ(n′)] U +λ2 qξ(n) ξ(n′) J~ R~ . { − − − }h i { − }h · (4.21)i 4.1 Representations in an Angular Momentum Basis 57

Provided Eq. (4.18) holds, the determinant condition for a nontrivial solution is

0 = [2]2 [ξ(k′)2 [2]ξ(k′)ξ(k′)+ ξ(k)2] [ξ(n′)2 [2]ξ(n′)ξ(n′)+ ξ(n)2] { − − − } = [2]2[ξ(k′)2 [2]ξ(k′)ξ(k′)+ ξ(k)2]+ λ2 − = λ4 k + 1 k′ k 1 k′ k + k′ + 1 k + k′ + 3 , (4.22) 2 − − 2 − 2 2 which is fulﬁlled precisely for k′ = k1 . We conclude that the possible transitions ± 2 of the quantum numbers n and k are n n 1 and k k 1 . → ± 2 → ± 2 4.1.4 Dependence on Total Angular Momentum Eq. (4.21) establishes a correspondence between the reduced matrix elements of J~ R~ and U. With Eq. (4.11) we get for j > 0 · ξ(n) qξ(n′) [2] j, n′,k′ U j, n, k j, n′,k′ R~ j, n, k = − ξ(j) h k k i h k k i ξ(k) qξ(k′) − λ2 [2j + 2][2j] − =: A (n′,k′, n, k, j) j, n′,k′ U j, n, k . (4.23) 1 h k k p i The reduced matrix elements of Eq. (3.35a) between j +1, n′,k′ and j, n, k , h k k i j 1, n′,k′ and j, n, k yield h − k k i j +1, n′,k′ R~ j, n, k = A (n′,k′, n, k, j) j, n′,k′ U j, n, k (4.24a) h k k i 2 h k k i j 1, n′,k′ R~ j, n, k = A (n′,k′, n, k, j) j, n′,k′ U j, n, k , (4.24b) h − k k i 3 h k k i where

(λ [2j] A q) j +1, n′,k′ P~ j, n′,k′ [2j+2] 1 − h k k i A2 := (4.25a) q[4] [2j+4] m [2]ξ(n) ξ(n′) + λ j +1, n′,k′ P~ j +1, n′,k′ − [2] [2j+2] h k k i q (λ [2j+2] A + q) j 1, n′,k′ P~ j, n′,k′ − [2j] 1 h − k k i A3 := . (4.25b) q[4] [2j−4] m [2]ξ(n) m ξ(n′) λ j 1, n′,k′ P~ j 1, n′,k′ − [2] − [2j] h − k k − i q This again can be used to calculate the reduced matrix elements of Eq. (3.35g) between j +1, n′,k′ and j, n, k h k k i j +1, n′,k′ U j +1,n,k = A (n′,k′, n, k, j) j, n′,k′ U j, n, k , (4.26) h k k i 4 h k k i where

A := [4] λ2 [2j] A j +1, n′,k′ P~ j, n′,k′ λ2A mqξ(n′) 4 [2] − [2j+2] 1 h k k i − 2 n q [2j+4] j +1, n′,k′ P~ j +1, n′,k′ [2]−1 j +1,n,k P~ j, n, k −1 . (4.27) − [2j+2] h k k i h k k i q o 58 4. Massive Spin Representations

The calculation of the auxiliary functions A1, A2, A3, and A4 is elementary but lengthy.2 The results can be written most compactly introducing the functions u(n′,k′,n,k) and v(n′,k′,n,k) by

u(n + ∆n, k + ∆k,n,k) := ∆n (2n +1)+∆k (2k + 1) (4.28) v(n + ∆n, k + ∆k,n,k) := ∆n (2n + 1) ∆k (2k + 1) , − for ∆n, ∆k = 1 , that is, ± 2 n′ = n 1 , k′ = k 1 u = n k 1 , v = n + k − 2 − 2 ⇒ − − − − n′ = n 1 , k′ = k + 1 u = n + k , v = n k 1 − 2 2 ⇒ − − − − (4.29) n′ = n + 1 , k′ = k 1 u = n k , v = n + k +1 2 − 2 ⇒ − n′ = n + 1 , k′ = k + 1 u = n + k +1 , v = n k 2 2 ⇒ − Using u we can write A4 as

′ ′ ′ ′ [j + u + 2][j u + 1] A5(n ,k , n, k, j + 1) A4(n ,k , n, k, j)= − = ′ ′ , (4.30) [j + u + 1][j u] A5(n ,k , n, k, j) p − where p

A (n′,k′, n, k, j) := [j + u + 1][j u] . (4.31) 5 − Deﬁning p

′ ′ ′ ′ j, n ,k U j, n, k n ,k U n, k := h ′ k′ k i , (4.32) h k k i A5(n ,k , n, k, j) Eq. (4.26) tells us by induction that n′,k′ U n, k does not depend on j. With h k k i Eqs. (4.23) and (4.24) we conclude that the j-dependence of all reduced matrix elements can be absorbed in reduction coeﬃcients according to

′ ′ ′ 0 ′ ′ ′ ′ ′ j , n ,k U j, n, k = Bq (j , n ,k j, n, k) n ,k U n, k h k k i | h k k i (4.33) j′, n′,k′ R~ j, n, k = B1(j′, n′,k′ j, n, k) n′,k′ U n, k , h k k i q | h k k i if we deﬁne the coeﬃcients as

′ ′ ′ 0 ′ ′ ′ A5(n ,k , n, k, j) , j = j > 0 Bq (j , n ,k j, n, k) := (4.34a) | (0 , else ′ ′ ′ ′ ′ A3(n ,k , n, k, j)A5(n ,k , n, k, j) , j = j 1 ′ ′ ′ ′ ′ − 1 ′ ′ ′ A1(n ,k , n, k, j)A5(n ,k , n, k, j) , j = j > 0 Bq (j , n ,k j, n, k) := ′ ′ ′ ′ ′ (4.34b) | A2(n ,k , n, k, j)A5(n ,k , n, k, j) , j = j +1  0 , else .  2The calculation of the auxiliary functions has been done by computer algebra [58]. 4.1 Representations in an Angular Momentum Basis 59

Explicitly, the formulas for the B-coeﬃcients are

0 ′ ′ ′ B (j , n ,k j, n, k)= δ ′ [j + u + 1][j u] q | jj − qp−j [2][j + v][j v][j u][j u 1] B1(j 1, n′,k′ j, n, k)= − − − − q − | − λ [2j][2j 1] p − 1 ′ ′ (j+1) p −(j+1) [j + u + 1][j u] Bq (j, n ,k j, n, k)= (q [j v] q [j + v]) − | − − − pλ [2j + 2][2j] j+1 1 ′ ′ q [2][j + v + 1][j v + 1][j +pu + 2][j + u + 1] Bq (j +1, n ,k j, n, k)= − , | − p λ [2j + 3][2j + 2] (4.35) p which can be written more compactly as

Bα(j′, n′,k′ j, n, k)= q | q−j [j′ u + 1][j′ u] , j′ = j 1 − − − −α ′ ′ ′ ′ ( λ) Cq(α, j , j 0,v,v) [j + u + 1][j u] , j = j (4.36) − | ×  p −  qj+1 [j′ + u + 1][j′ + u] , j′ = j +1 . −p  p 4.1.5 Dependence on the other Quantum Numbers Using the B-coeﬃcients, equations in the reduced matrix elements of R, U can be reduced further to equations in the double reduced matrix elements n′,k′ U n, k as deﬁned in Eq. (4.32). We start by taking the matrix elements h k k i of the RR-relations (2.70a), U 2 λ2 (R~ R~ ) = 1 and RAU URA = 0 between j, n, k and j, n, k . We obtain− · − h k k i ′ ′ ′ ′ ′ ′ A6(n ,k , n, k, j) n, k U n ,k n ,k U n, k = 1 (4.37a) ′ ′ h k k ih k k i nX,k ′ ′ ′ ′ ′ ′ A7(n ,k , n, k, j) n, k U n ,k n ,k U n, k =0 , (4.37b) ′ ′ h k k ih k k i nX,k where the summation indices run through n′ = n 1 , k′ = k 1 and ± 2 ± 2 A (n′,k′, n, k, j) := B0(j, n, k j, n′,k′)B0(j, n′,k′ j, n, k) 6 q | q | j+1 ′ 2 j′−j [2j +1] 1 ′ ′ ′ 1 ′ ′ ′ λ ( 1) [2j+1] Bq (j, n, k j , n ,k )Bq (j , n ,k j, n, k) (4.38a) − ′ − | | j X=j−1 q λ A (n′,k′, n, k, j) := [2j + 2][2j] B1(j, n, k j, n′,k′)B0(j, n′,k′ j, n, k) 7 −[2] q | q | p n B0(j, n, k j, n′,k′)B1(j, n′,k′ j, n, k) . (4.38b) − q | q | o 60 4. Massive Spin Representations

The values of these coeﬃcients are ′ ′ ′ ′ A6(n ,k , n, k, j)=4∆k ∆n [2][2k + 1][2n + 1] (4.39) A (n′,k′, n, k, j) = [2v][j + u + 1][j u]= λ−2[2][2v] ξ(j) ξ(u) . 7 − − Eq. (4.37b) must hold for all values of j, which turns out to lead to two independent equations. Thus, Eqs. (4.37) form a system of three independent equations in four unknowns of the type n, k U n′,k′ n′,k′ U n, k . Eliminating two unknowns in each equation we canh interpretk k themih ask recursionk i relations

ρ(µ, ν)= ρ(µ, ν 1) + [2ν + 2] (4.40a) − ω(µ, ν)= ω(µ +1, ν) + [2µ] (4.40b) ω(µ +1, ν)= ρ(µ, ν) + [ν + µ + 2][ν µ + 1] (4.40c) − − where we use the abbreviations µ := k n, ν := k + n and − ρ(µ, ν) := [2]2[2k + 2][2k + 1][2n + 2][2n + 1] n, k U n + 1 ,k + 1 n + 1 ,k + 1 U n, k (4.41a) ×h k k 2 2 ih 2 2 k k i ω(µ, ν) := [2]2[2k + 1][2k][2n + 2][2n + 1] n, k U n + 1 ,k 1 n + 1 ,k 1 U n, k . (4.41b) ×h k k 2 − 2 ih 2 − 2 k k i In order to determine the initial conditions, we recall Eq. (4.4) which tells us that n = 0 implies k = s. Hence, matrix elements involving states with n = 0 and k = s have to vanish, in particular 6 ρ(s,s 1)=0 . (4.42) − The solution of recursion relation (4.40a) with this initial value is

ν ρ(s, ν)= [2ν′ + 2] = [ν + s + 2][ν s + 1] (4.43) ′ − νX=s b ′ where we used i′=a[2i + c] = [a + b + c][b a + 1]. Inserting this result in Eq. (4.40c) yields ω(s +1, ν) = 0. The solution− of Eq. (4.40b) with this initial value is P s ω(µ, ν)= [2µ′] = [µ + s][s µ + 1] . (4.44) ′ − µX=µ Inserting this again in Eq. (4.40c) results in ω(µ, ν)= ω(µ) = [µ + s][s µ + 1] − (4.45) ρ(µ, ν)= ρ(ν) = [ν + s + 2][ν s + 1] − 4.1 Representations in an Angular Momentum Basis 61 for s µ s + 1 and s 1 ν. At the border of this half-closed strip in µν−-space≤ ρ≤and ω vanish,− so there≤ are no transitions to the outside. For an irreducible representation we must not have two disconnected regions, hence, ρ and ω must vanish outside this strip. The allowed quantum numbers form a strip in nk-space given by

µ = k n s , ν = n + k s . (4.46) | | | − | ≤ ≥ To derive from Eq. (4.45) formulas for the matrix elements we need to take the RS-relations (2.70c) into account. We begin with the matrix elements of UV = V U between j, n + 1 ,k + 1 and j, n 1 ,k 1 using the conjugation h 2 2 k k − 2 − 2 i U ∗ = V to obtain

n + 1 ,k + 1 U n, k n 1 ,k 1 U n, k = h 2 2 k k ih − 2 − 2 k k i n, k U n + 1 ,k + 1 n, k U n 1 ,k 1 , (4.47) h k k 2 2 ih k k − 2 − 2 i which can be written as

µ, ν 1 U µ, ν µ, ν U µ, ν +1 h − k k i = h k k i . (4.48) µ, ν U µ, ν 1 µ, ν +1 U µ, ν h k k − i h k k i with µ := k n, ν := k +n as above. Reading this as recursion relation, it follows that −

µ, ν 1 U µ, ν = α µ, ν U µ, ν 1 , (4.49a) h − k k i µh k k − i where the yet to be determined number αµ may depend on µ but not on ν. Taking ′ ′ 1 1 1 1 the matrix elements of UU = U U between j, n + 2 ,k + 2 and j, n 2 ,k 2 , it follows analogously that h k k − − i

µ, ν U µ 1, ν = β µ 1, ν U µ, ν , (4.49b) h k k − i ν h − k k i with βν independent of µ. Next, we take the diagonal matrix elements of W = UV + q2λ2(R~ S~) as in · Eq. (2.77) using the conjugation relations (A.19e) to obtain

′ ′ ′ ′ 2 A8(n ,k , n, k, j) n, k U n ,k = ξ(j) , (4.50) ′ ′ |h k k i| nX,k where

j+1 ′ ′ 0 ′ ′ 2 2 2 1 ′ ′ ′ 2 A8(n ,k , n, k, j) := Bq (j, n, k j, n ,k ) + q λ Bq (j, n, k j , n ,k ) . | | | ′ | | | j X=j−1 (4.51) 62 4. Massive Spin Representations

Eq. (4.50) must hold for all possible values of j, thus yielding two independent equations from which we can derive

[2]−2[µ + ν + 1]−1 = q−2µ[ν µ] µ, ν U µ, ν 1 2 − |h k k − i| + q−2(ν+1)[ν µ + 2] µ, ν U µ 1, ν 2 . (4.52) − |h k k − i| Relations (4.46) tell us that the ﬁrst term on the right hand side vanishes for ν = s while the second vanishes for µ = s, that is, − q−2s s, ν U s, ν 1 2 = |h k k − i| [2]2[ν s + 1][ν + s] − (4.53) q2(s+1) µ,s U µ 1,s 2 = . |h k k − i| [2]2[µ + s + 1][s µ + 2] − If we compare this with ρ( s, ν 1) and ω(µ,s) as computed in Eqs. (4.45), we ﬁnd − −

2s 2(s+1) αµ = q , βν = q . (4.54)

With this result Eqs. (4.45) can be written as formulas for the squares of matrix elements. For example,

[µ + s][s µ +1] = ω(µ, ν) − = [2]2[2k + 1][2k][2n + 2][2n + 1] n, k U n + 1 ,k 1 n + 1 ,k 1 U n, k h k k 2 − 2 ih 2 − 2 k k i = q2(s+1)[2]2[2k + 1][2k][2n + 2][2n + 1] n + 1 ,k 1 U n, k 2 . (4.55) |h 2 − 2 k k i| This is an equation for the absolute value of the double reduced matrix elements. In fact, none of the commutation relations of the q-Poincar´ealgebra gives us a condition on the phase of the reduced matrix elements, that is, the phase can be chosen arbitrarily. We choose it, such that

−2(s+1) 1 1 q [s + k n][s k + n + 1] n + 2 ,k 2 U n, k = − − . (4.56) h − k k i [2] [2pk + 1][2k][2n + 2][2n + 1] Analogously, we determine the otherp matrix elements. The end result is

′ ′ ′ q2(n−n )s+(n −k −n+k) [s + u + 1][s u] n′,k′ U n, k = − . (4.57) h k k i ′ 3 ′ 1 ′ 3 ′ 1 [2] [k + k + 2 ][k + k +p2 ][n + n + 2 ][n + n + 2 ] q Summary We summarize the results for the reduced matrix elements. As before, the abbreviations u and v as deﬁned in Eq. (4.28) are being used. The relation between the reduced and the ordinary matrix elements is given by Eqs. (4.9) and (4.10). 4.1 Representations in an Angular Momentum Basis 63

′ ′ ′ ~ −1 ′ ′ ′ j , n ,k J j, n, k = [2] δjj δnn δkk [2j + 2][2j] (4.58a) h k k i − ′ ′ j 1, n ,k P~ j, n, k = mλδ ′ δ ′ p h − k k i − nn kk [k + n + j + 1][k + n j + 1][k n + j][n k + j] − − − (4.58b) × [2][2j][2j 1] p − ′ ′ j, n ,k P~ j, n, k = mλδ ′ δ ′ p h k k i nn kk [k + n + j + 2][j k + n] [k + n j][j + k n] − − − − (4.58c) × [2] [2j + 2][2j]

j +1,n,k P~ j, n, k = mλδ ′ δ ′ p h k k i nn kk [k + n + j + 2][k + n j][k n + j + 1][n k + j + 1] − − − (4.58d) × p [2][2j + 3][2j + 2] ′ ′ ′ 2(n−n′)s+(n′−k′−n+k) j , n ,k U j, n, k = δ ′ q p h k k i jj [j + u + 1][j u][s + u + 1][s u] − − . (4.58e) × ′ 3 ′ 1 ′ 3 ′ 1 [2] [k p+ k + 2 ][k + k + 2 ][n + n + 2 ][n + n + 2 ] q j′, n′,k′ R~ j, n, k = h k k i ′ ′ ′ q2(n−n )s+(n −k −n+k) [s + u + 1][s u] − ′ 3 ′ 1 ′ 3 ′ 1 λ[2] [k + k + 2 ][k + k +p2 ][n + n + 2 ][n + n + 2 ] q q−j [j′ u + 1][j′ u] , j′ = j 1 − − − − ′ ′ ′ ′ Cq(1, j , j 0,v,v) [j + u + 1][j u] , j = j (4.58f) × | × − p −  j+1 ′ ′ ′ q p [j + u + 1][j + u] , j = j +1 . ′ ′ ′ 2(n′−n)s+(n−k−n′+k′) j , n ,k V j, n, k = δjj′ q  p h k k i [j + u + 1][j u][s + u + 1][s u] − − . (4.58g) × ′ 3 ′ 1 ′ 3 ′ 1 [2] [k p+ k + 2 ][k + k + 2 ][n + n + 2 ][n + n + 2 ] q j′, n′,k′ S~ j, n, k = h k k i ′ ′ ′ q2(n −n)s+(n−k−n +k ) [s + u + 1][s u] − ′ 3 ′ 1 ′ 3 ′ 1 λ[2] [k + k + 2 ][k + k +p2 ][n + n + 2 ][n + n + 2 ] q qj [j′ u + 1][j′ u] , j′ = j 1 − − − − ′ ′ ′ ′ Cq(1, j , j 0, v, v) [j + u + 1][j u] , j = j (4.58h) × | − − × − p −  −(j+1) ′ ′ ′ q p [j + u + 1][j + u] , j = j +1  p 64 4. Massive Spin Representations

4.2 Representations by Induction We want to describe brieﬂy how representations of the q-Poincar´ealgebra can be constructed using the method of induced representations.

4.2.1 The Method of Induced Representations of Algebras Let us assume that we do have an irreducible representation of the undeformed Poincaréalgebra on a Hilbert space , P H σ : . (4.59) P ⊗ H −→ H Let the situation be as in Sec. 3.3.1, where we denoted by a momentum Hp eigenspace and by p its stabilizer (little algebra). By definition, the restriction of σ to defines representationsK on translations and the little algebra by Hp T Kp

χp : R , where σ(t ψp )= χp(t) ψp T −→ ⊗| i | i (4.60) ρ : , ρ(k ψ )= σ(k ψ ) Kp ⊗ Hp −→ Hp ⊗| pi ⊗| pi for all ψp p. Together, χp and ρ deﬁne a representation of ⋊ p on p. Let us| assumei ∈ H for a moment that we did not know about σ but wereT K given onlyH χp and ρ. There is a generic method to extend a representation of an subalgebra to a representation of the whole algebra.

Deﬁnition 8. Let be an algebra, a subalgebra and V a left -module. Then A S S the tensor product of and V over , S V becomes a left -module by left multiplication. It is calledA the moduleS (orA representation ⊗ ) inducedA by V . Explicitly, S V is the vector space V (ordinary tensor product over the complex numbers),A ⊗ divided by the relationsA ⊗

as v = a sv , for all a ,s , v V, (4.61) ⊗ ⊗ ∈A ∈ S ∈ with the left -action deﬁned by A a′(a v)= a′a v (4.62) ⊗ ⊗ and linear extension.

For given χ , , ρ, and , the induced representation acts on the tensor p Hp Kp product

⋊ =( ⋊ ) ⋊ = . (4.63) P ⊗T Kp Hp T L ⊗T Kp Hp ∼ L ⊗Kp Hp While this construction may look somewhat abstract, its great practical value lies in the following 4.2 Representations by Induction 65

Theorem 2. Let = ⋊ be the Poincaréalgebra, χp a one dimensional representation of P, T= kL χ ([k, t]) = 0 for all t the according T Kp { ∈L| p ∈T} little algebra, and ρ an irreducible representation of p on the finite vector space . With the action defined by χ and ρ the spaceK becomes a left ⋊ - Hp p Hp T Kp module. Then the induced representation T ⋊Kp p is irreducible. Furthermore, all irreducible representations of are ofP⊗ this formH [54,55]. P This means that all we have to do in order to construct the irreducible representations of is P 1. determine the little algebras,

2. construct the irreducible representations of the little algebras,

3. induce these representations.

Using the Lie group version of this method, Wigner [1] was the ﬁrst to construct all irreducible representations of the Poincar´egroup (see also [59]). Theorem 2 cannot be generalized to Hopf semidirect products but in very special cases [56, 60, 61]. The method of induced representations, however, works for any algebra.

4.2.2 Induced Representations of the q-Poincar´eAlgebra

We will deal only with the massive case, p =(pµ)=(m, 0, 0, 0) = χp(Pµ), where we have = (su ), as calculated in Sec. 3.3.2. Let Dj be an irreducible Kp Uq 2 q(su2)-module. Recall (p. 38) the deﬁnition of the q-Poincar´ealgebra q = U ⋊ (sl (C)). In the quantum double form (Sec. 2.3) the q-Lorentz algebraP Mqq Uq 2 is (sl (C)) = SU (2)op (su ) as vector space. We conclude that the induced Uq 2 ∼ q ⊗Uq 2 representation of Dj acts on the vector space

j j ⋊ D = [ ⋊ (sl (C))] ⋊ D Pq ⊗Mq Uq (su2) Mq Uq 2 ⊗Mq Uq(su2) = (sl (C)) Dj ∼ Uq 2 ⊗Uq (su2) = (SU (2)op (su )) Dj ∼ q ⊗ Uq 2 ⊗Uq (su2) = SU (2)op Dj. (4.64) ∼ q ⊗ j ′ op Let em be a basis of D . The action of some boost h SUq(2) on h em SU (2)op Dj is simply given by left multiplication ∈ ⊗ ∈ q ⊗ h′(h e )= h′h e . (4.65) ⊗ m ⊗ m For the action of a rotation l (su ) we have to commute lh using Eq. (2.57) ∈ Uq 2 and let l act on em

j m′ l(h e )= S(l ), h l , h (h e ′ ρ (l ) ) . (4.66) ⊗ m h (1) (1)ih (3) (3)i (2) ⊗ m (2) m 66 4. Massive Spin Representations

Finally, for the action of P we must use Eq. (3.30), µ ∈Mq −1 µ′ P (h ψ)= p ′ Λ(S h ) (h e ) , (4.67) µ ⊗ µ (1) µ (2) ⊗ m where pµ = χp(Pµ) are the momentum eigenvalues. We can equip this representation with a scalar product using the Haar measure of SUq(2) ( [15], see also [62], pp. 111-117). An orthogonal basis is provided by the Peter-Weyl theorem ( [62], pp. 106-111). Chapter 5 Free Wave Equations

5.1 General Wave Equations 5.1.1 Wave Equations by Representation Theory On the way from free theories to theories with interaction we need to leave the mass shell. The space of on-shell states is clearly too small as to allow for interactions where energy and momentum can be transfered from one sort of particle onto another. Moreover, we need a way to describe several particle types and their coupling in one common formalism. These issues are resolved by introducing Lorentz spinor wave functions, that is, tensor products of the algebra of functions on spacetime with a ﬁnite vector space containing the spin degrees of freedom, the whole space carrying a tensor representation of the Lorentz symmetry. The additional mathematical structure we need to describe coupling is provided by the multiplication within the algebra of space functions. This structure is equally present in the undeformed as in the deformed case. Using such Lorentz spinors has some consequences that have to be dealt with:

(a) The Lorentz spinor representations cannot be irreducible. Otherwise they would have to be on shell and the spinor degrees of freedom would have to carry a representation of the little algebra.

(b) The Lorentz spinor representations cannot be unitary since the spin degrees of freedom carry a ﬁnite representation of the non-compact Lorentz algebra.

The solution to these problems are:

(a) We consider only an irreducible subrepresentation to be the space of physical states. This subrepresentation is described as kernel of a linear operator A, that is, we demand all physical states ψ to satisfy the wave equation Aψ = 0.

(b) We introduce a non-degenerate but indeﬁnite pseudo scalar product, such that the spinor representation becomes a -representation with respect to ∗ the corresponding pseudo adjoint. This amounts to introducing a new conjugation j on states and operators. 68 5. Free Wave Equations

For ker A to be a subrepresentation, the operator must satisfy

Aψ =0 Ahψ = 0 (5.1) ⇒ for all q-Poincar´etransformations h. Depending on the particle type under consideration we might include charge and parity transformations. A is not unique since the wave equations for A and A′ must be considered equivalent as long as their solutions are the same, ker(A) = ker(A′). Ideally, the operator A is a projector, A = P, with P2 = P, P∗ = P. Condi- tion (5.1) is then equivalent to

[P, h] = 0 (5.2) for all q-Poincarétransformations h. Whether the wave equation is written with a projection is a matter of convenience. The Dirac equation is commonly written with such a projection which is determined uniquely (up to complement) by condition (5.2). For the Maxwell equations a projection can be found but yields a second order differential equation. For this reason, the Maxwell equations are commonly described by a more general operator A, which leads to a first order equation. So far, all considerations pertain equally to the deformed as to the undeformed case.

5.1.2 q-Lorentz Spinors We deﬁne a general, single particle q-Lorentz spinor wave function as element of the tensor product q of a ﬁnite vector space holding the spin degrees of freedom and the spaceS⊗M of q-Minkowski space functionsS (Sec. 3.1.1). Mq Let ek be a basis of transforming under a q-Lorentz transformation { } S i h q(sl2(C)) as h ⊲ ej = ei ρ(h) j, where ρ : q(sl2(C)) End( ) is the representation∈ U map. Any spinor ψ can be written asU → S

ψ = e ψj , (5.3) j ⊗ j where j is summed over and the ψ are elements of q. The total action of h (sl (C)) on a spinor is M ∈ Uq 2 hψ =(h ⊲ e ) (h ⊲ ψj)= e ρ(h )i (h ⊲ ψj) . (5.4) (1) j ⊗ (2) i ⊗ (1) j (2)

This tells us that, if we want to work directly with the q-valued components ψj, the action of h is M

i i j hψ = ρ(h(1)) j(h(2) ⊲ ψ ) . (5.5)

Do not confuse the total action hψi with the action of h on each component of ψi denoted by h ⊲ ψi. The transformation of ψi can easily be generalized to the 5.1 General Wave Equations 69 case where carries a tensor representation of two ﬁnite representations, that is, we have spinorsS with two or more indices

ij i ′ j i′j′ hψ = ρ(h(1)) i′ ρ (h(2)) j′ (h(3) ⊲ ψ ) , (5.6) where ρ and ρ′ are the representation maps of the first and second index, respectively. Furthermore, we can derive spinors by the action of tensor operators: Let T i j ′ be a upper left ρ-tensor operator and ψ = ej ψ a ρ -spinor field. Any operator T i can be written as T i = Ai Bi End(⊗ ) End( ) such that the action k k ⊗ k ∈ S ⊗ Mq of T i becomes P i i j i j′ i j ij T ψ = e ρ(A ) ′ B ⊲ ψ = e (T ψ ) =: e φ . (5.7) j ⊗ k j k j ⊗ j ⊗ Xk How does this new array of wave functions φij = T iψj transform under q-Lorentz transformations? Letting act h from the left, we find

ij j ij′ hφ = ρ(h(1)) j′ (h(2) ⊲φ ) , (5.8) that is, h acts only on the index that came from the wave functions ψj. However, if we transform φij by transforming ψj inside, we ﬁnd

i j i j −1 i j T (hψ )=(T h)ψ = h(2)[adLS (h(1)) ⊲ T ]ψ i i′ j i i′j = ρ(h(1)) i′ h(2)T ψ = ρ(h(1)) i′ h(2)φ i ′ j i′j′ = ρ(h(1)) i′ ρ (h(2)) j′ (h(3) ⊲φ ) . (5.9)

In other words, if ψj is transformed φij = T iψj will transform as a ρ ρ′- spinor. Note, that for the last calculation the order in the tensor product⊗ is essential. This reasoning would not have worked out as nicely if weS ha ⊗d Mq constructed the spinor space as q . Chief examples of this construction are the gauge term P µφ of the vectorM potential⊗ S Aµ, or the derivatives of the vector potential P µAν which are used to construct the electromagnetic ﬁeld strength tensor F µν . We have not said yet how the momenta P µ act on q-Lorentz spinors. One might be tempted to assume that they act on the wave function part only, that is, as 1 P µ on the tensor product. However, this is not possible, as in general 1 P µ ⊗is no 4-vector operator and thus cannot represent 4-momentum. We can ⊗ turn 1 Pµ into a vector operator, though, by twisting 1 Pµ with an -matrix of the q⊗-Lorentz algebra, ⊗ R

µ −1 µ Λ µ µ′ P := (1 P ) =(L ) ′ P , (5.10) R ⊗ R + µ ⊗ with the L-matrix for the 4-vector representation as deﬁned in Eq. (2.43). Of the two universal -matrices of the q-Lorentz algebra we opt for , because R RI 70 5. Free Wave Equations only then the twisting is compatible with the -structure. The momenta act on a ρ-spinor as ∗

i ′ µ i Λ µ ′ µ j P ψ = ρ (LI+) µ j (P ⊲ ψ ) , (5.11) where the L-matrix has been calculated in Eq. (3.80). The action of P µ on each component of ψj can be viewed as derivation within the algebra of q-Minkowski space functions . The q-derivation operators are Mq ∂µ := 1 iP µ . (5.12) ⊗ Now we can interpret an operator linear in the momenta as q-diﬀerential operator. If C = C 1 are operators that act on the spinor indices only, µ µ ⊗ ′ ′ µ Λ µ ′ µ ˜ ′ µ i CµP = Cµ ρ (LI+) µ ∂ = Cµ ∂ , (5.13) where

˜ ′ Λ µ ′ Cµ := Cµ ρ (LI+) µ (5.14)

µ′ such that C˜µ′ still acts on the spinor index only, while ∂ acts componentwise, ν so the two operators commute [C˜µ, ∂ ] = 0. It remains to calculate the transformation Cµ C˜µ for particular representations ρ. Finally, we remark that → µ −1 µ µ for the mass Casimir we have PµP = (1 PµP ) = 1 PµP , hence, P P µ = ∂ ∂µ. This means, that mass irreducibilityR ⊗ forR a spinor⊗ is the same as µ − µ mass irreducibility for each component of the spinor.

5.1.3 Conjugate Spinors One of the effects of using Lorentz spinors is that the underlying representations can no longer be unitary, since there are no unitary finite representations of the non-compact Lorentz algebra. However, we can introduce non-degenerate but indefinite bilinear forms playing the role of the scalar product. With respect to these pseudo scalar products the spinors carry -representations, that is, the - ∗ ∗ operation on the algebra side is the same as the pseudo adjoint on the operator side. The problem of non-unitarity arises from the finiteness of the spin part, , S within the space of spinor wave functions q, so we can assume that the wave function part does carry a -representation.S⊗M It is then sufficient to redefine Mq ∗ the scalar product on only. Consider a D(j,0)-representation of (sl (C)) with S Uq 2 orthonormal basis em and the canonical scalar product em en = δmn. We want to define a pseudo{ } scalar product by h | i

(e e ) := A such that (e (g h) ⊲ e )=((g h)∗ ⊲ e e ) (5.15) m| n mn m| ⊗ n ⊗ m| n 5.1 General Wave Equations 71

for any g h q(sl2(C)). For a pseudo scalar product we must suppose Amn to be a non-degenerate,⊗ ∈ U hermitian, but not necessarily positive deﬁnite matrix. Inserting the deﬁnition of the pseudo scalar product, the pseudo-unitarity condition (5.15) reads

j n′ j n′ (e (g h) ⊲ e )=(e e ′ ρ (g) ε(h)) = A ′ ρ (g) ε(h) m| ⊗ n m| n n mn n ! ∗ ∗ j ∗ m′ =((g h) ⊲ e e )=(e ′ ε(g )ρ (h ) e ) ⊗ m| n m m| n ∗ j ∗ m′ j m = Am′n ε(g )ρ (h ) m = Am′n ε(g)ρ (h) m′ , (5.16) where we have used the deﬁnition (1.52) of (g h)∗ observing that ε( ) = ⊗ R[1] R[2] 1. Traditionally, the scalar product is not described by a matrix Amn but by introducing a conjugate spinor basis e¯ demanding { m} (e e )= e¯ e e¯ = e ′ A ′ . (5.17) m| n h m| ni ⇒ m m m m Using (5.16) the conjugate basis turns out to transform as

j m′ j n′ (g h) ⊲ e¯ = e ′ ρ (g) ε(h) A = e ′ A ′ ′ ε(g)ρ (h) ⊗ n m m mn m m n n j n′ =e ¯n′ ε(g)ρ (h) n , (5.18)

(0,j) (j,0) that is,e ¯m ought to transform according to a D -representation. D and (0,j) D being inequivalent representations, the conjugate basise ¯m cannot be expressed as a linear combination of the original basis vectors em. In order to allow for a conjugate spinor basis we must consider a representation that contains both, D(j,0) and D(0,j), and thus at least their direct sum D(j,0) D(0,j) as subrepresentation. ⊕ So far it seems that everything is almost trivially analogous to the undeformed case. It is not. If we consider irreducible representations of mixed chirality, D(i,j), we find that the appearance of the -matrix in (g h)∗ makes it impossible to R (j,0) ⊗ 0 define conjugate spinors. It only works for D , because ρ = ε and ε( [1]) [2] = 1. Fortunately, we do have conjugate spinors for the most interesting cases:R R Dirac ( 1 ,0) (0, 1 ) (1,0) (0,1) spinors (D 2 D 2 ) and the Maxwell tensor (D D ). For these cases ⊕ ⊕ everything is analogous to the undeformed case. (j,0) (0,j) L Let us consider a D D representation with basis em for the left chiral (j,0) ⊕ R (0,j) { } subrepresentation D and the basis em for D . We define the conjugate L R R L { } basis by em := em and em = em. Let us call the parity operator that exchanges the left and right chiral part. Its matrix representationP in the basis eL , eR is { m m} 0 1 = , (5.19) Pmn 1 0 where 1 is the (2j + 1)-dimensional unit matrix. This is the matrix that repre- sents our new pseudo scalar product as a bilinear form. The pseudo Hermitian conjugate of some operator A can now be written as j(A) := A† , (5.20) P P 72 5. Free Wave Equations which is an involution because = † and an algebra anti-homomorphism because = −1. P P P P We apply this result to the whole space of spinor wave functions q. Let us assume that the scalar product of two wave functions f,g S⊗Mcan be ∈ Mq written (at least formally) as some sort of integral f g = f ∗g. The pseudo scalar product of two D(j,0) D(0,j) spinors ψ, φ becomesh | i ⊕ R (ψ φ)=(e ψm e φn)=(e e ) ψm φn | m ⊗ | n ⊗ m| n h | i = (ψm)∗ φn = ψ¯nφn , (5.21) Pmn with the conjugate spinorR wave function definedR as

ψ¯n := (ψm)∗ . (5.22) Pmn To summarize, we have convinced ourselves that in the case of D(j,0) D(0,j) representations the conjugation of spinors, spinor wave functions and operators⊕ works exactly as in the undeformed case.

5.2 The q-Dirac Equation 5.2.1 The q-Dirac Equation in the Rest Frame

j In this section we consider q-Dirac spinors ψ = ej ψ with the spin part trans- ( 1 ,0) (0, 1 ) ⊗ forming according to a D 2 D 2 representation. We hope that we can write the projection onto an irreducible⊕ component of the space of q-Dirac spinors as expression which involves momenta only to first order terms, corresponding to a first order differential equation. The general expression for such a q-Dirac equation would be 1 Pψ := (m + γ P µ)ψ =0 , (5.23) 2m µ

j with γµ being some operators acting on ψ . We can already say that γµ must µ be a left 4-vector operator. If it were not, γµP would not be scalar and, hence, would not commute with the q-Lorentz transformations as required in Eq. (5.2). We consider here a massive q-Dirac spinor representation, so there is a rest frame (Sec. 3.3.2), that is, a set of states ψj, which the momenta act upon as P 0ψj = mψj, P Aψj = 0. We start the search for a projector P that reduces the q-Dirac representation by computing how it has to act on the rest frame, where we have

1 P0 = 2 (1 + γ0) , (5.24) the zero indicating that this is a projector within the rest frame only. We assume that we can realize the operator γ as 4 4-matrix that acts on the spin degrees 0 × 5.2 The q-Dirac Equation 73

of freedom only. This is not unreasonable, for if γµ is a set of matrices that form ( 1 ,0) (0, 1 ) a 4-vector operator in the D 2 D 2 representation then γ 1 will also ⊕ µ ⊗ be a 4-vector operator in the representation of spinor wave functions. So, let us assume we can write P = P 1 in block form as 0 0 ⊗ A B P = , (5.25) 0 CD where A, B, C, D are 2 2-matrices. × Recall that P0 must satisfy condition (5.2). This tells us in particular that P0 must commute with rotations, the symmetry of the rest frame. A rotation l is represented by

1 ρ 2 (l) 0 ρ(l)= 1 . (5.26) 0 ρ 2 (l) 1 Since the ρ 2 -representations of the rotations generate all 2 2-matrices (the q- × Pauli matrices are a basis), P0 will only commute with all rotations if A, B, C, D are numbers, that is, complex multiples of the unit matrix. 2 † Furthermore, P0 has to be a projector, P0 = P0, P0 = P0, and, as in the undeformed case, we require it to commute with the parity operator, [P , ] = 0. 0 P Altogether these conditions ﬁx P0 and hence γ0 uniquely to be 0 1 γ = , (5.27) 0 1 0 the same as in the undeformed case.

5.2.2 The q-Gamma Matrices and the q-Cliﬀord Algebra

If γ0 is to be a 4-vector operator, we have to deﬁne the other gamma matrices as in Eq. (3.36) by

− 1 −1 1 γ = ad ( q 2 λ [2] 2 c) ⊲γ − L − 0 1 −1 1 γ+ = adL(q 2 λ [2] 2 b) ⊲γ0 (5.28) γ = ad (λ−1 (d a)) ⊲γ , 3 L − 0 where the adjoint action is understood with respect to the q-Dirac representation. To compute this, explicitly, we have to calculate the representations of the boosts ﬁrst.

1 1 ρ 2 (K 2 ) 0 0 0 ρ(a)= 1 − 1 , ρ(b)= − 1 1 − 1 (5.29a) 0 ρ 2 (K 2 ) 0 q 2 λρ 2 (K 2 E) 1 1 1 1 − 1 q 2 λρ 2 (FK 2 ) 0 ρ 2 (K 2 ) 0 ρ(c)= − , ρ(d)= 1 1 (5.29b) 0 0 0 ρ 2 (K 2 ) 74 5. Free Wave Equations

To demonstrate the simplicity of the technique of boosting, let us demonstrate it with an example.

1 −1 1 1 −1 1 γ+ = adL(q 2 λ [2] 2 b) ⊲γ0 = q 2 λ [2] 2 [ρ(b)γ0ρ(a) qρ(a)γ0ρ(b)] 1 − 1 0 0 0 ρ 2 (E) 0 q σ+ 2 = [2] 1 − 1 1 q = −1 (5.30) ρ 2 (K 2 EK 2 ) 0 − 0 0 q σ+ 0 −

Here, σ+ is the q-Pauli matrix (Sec. 3.1.2). After doing the other calculations we get

0 1 0 q σ γ = , γ = A , (5.31) 0 1 0 A q−1σ 0 − A where A runs as usual through , +, 3 . {− } This result can be easily generalized to higher spin massive particles. All we (j,0) (0,j) 1 j have to do for a massive D D -spinor is to replace ρ 2 with ρ in the above calculations. The result are higher⊕ dimensional γ-matrices

0 1 0 q ρj(J ) γ(j) = , γ(j) = [2] A . (5.32) 0 1 0 A q−1ρj(J ) 0 − A Now we want to write the q-Dirac equation as q-diﬀerential equation. Towards this end we need to calculateγ ˜µ by formula (5.14). Using Eqs (A.68) and (A.70) we get for the q-Pauli matrices

1 1 ( 2 ,0) Λ A 2 (0, 2 ) Λ A −2 σA ρ (LI+) B = q σ˜B , σA ρ (LI+) B = q σ˜B , (5.33) where

1 −1 1 0 q 2 1 0 0 q 0 2 2 σ˜− = [2] , σ˜+ = [2] − 1 , σ˜3 = − (5.34) 0 0 q 2 0 0 q − with respect to the , + basis. We can write this more compactly as {− } 1 σ˜ = [2] ρ 2 (SJ ) . (5.35) A − A

In this sense the transformed q-Pauli matrices,σ ˜A, can be viewed as antipodes of the original ones. For the transformed q-gamma matrices we obtain

0 1 0 q−1 σ˜ γ˜ = , γ˜ = A , (5.36) 0 1 0 A qσ˜ 0 − A so the q-Dirac equation written as q-diﬀerential equation becomes

(m i˜γ ∂µ)ψ =0 . (5.37) − µ 5.2 The q-Dirac Equation 75

What commutation relations do the gamma matrices satisfy? Using Eqs. (5.36) and (A.39) we ﬁnd after some lengthy calculations

ba γ˜cγ˜d = ηdc +˜γaγ˜bPA dc , (5.38) where PA is the antisymmetric projector defined in Eq. (A.29). This is the q- deformation of the Clifford algebra. Using the relations of the q-Clifford algebra it can be shown that the square of q-Dirac operator is indeed the mass Casimir,

(˜γ ∂µ)2 = ∂ ∂µ = P P µ . (5.39) µ µ − µ As in the undeformed case we conclude that a solution ψ to the q-Dirac equation µ 2 satisﬁes automatically the mass shell condition PµP ψ = m ψ, and that the 1 µ operator P = 2m (m + γµP ) really is a projector. The q-Cliﬀord relations (5.38) can be written in equivalent but more familiar forms as

ba ab γãγ˜bPS dc = ηdc , orγ ˜cγ˜d +˜γaγ˜bRII dc = q[2]ηcd , (5.40) with the symmetrizer (A.29) and the R-matrix (A.66). One could have started directly from these relations trying to find matrices that satisfy them [36]. This approach has a number of disadvantages: a) It is computationally much more cumbersome than boosting γ0. b) The result is not unique, that is, we would get many solutions to the q-Clifford algebra not knowing which representations they belong to. c) Having determined a solutionγ ˜µ, the covariance of the q-Dirac equation remains unclear asγ ˜µ cannot be a 4-vector operator.

5.2.3 The Zero Mass Limit and the q-Weyl Equations

µ The zero mass limit of the q-Dirac equations, (m + γµP )ψ = 0, is formally

µ γµP ψ =0 , (5.41)

µ where γµ is deﬁned as in Eq. (5.32). The operator A := γµP is no longer a projection. For m 0 the wave equation decouples into two independent → ( 1 ,0) (0, 1 ) equations for a left handed D 2 -spinor ψL and a right handed D 2 -spinor ψR,

σ P Aψ = q−1P 0ψ , σ P Aψ = qP 0ψ , (5.42) A L L A R − R 1 the q-Weyl equations for massless left and right handed spin- 2 particles. Written as q-diﬀerential equation they become

σ˜ ∂Aψ = q∂0ψ , σ˜ ∂Aψ = q−1∂0ψ . (5.43) A L − L A R R The operator A inherits property (5.1) from the massive q-Dirac projector P, so Aψ = 0 is a viable wave equation. Let us see what it looks like in the momentum 76 5. Free Wave Equations

eigenspace p for the momentum eigenvalues p =(p0,p−,p+,p3)=(k, 0, 0,k) for some real parameterH k (Sec. 3.3.2). On this subspace A acts as 0 0 q 0 0 1 qσ3 0 0 00 A Hp = k −1 − = k[2]   . (5.44) | 1+ q σ3 0 0 0 00 0 q−1 0 0     The kernel of this operator is 2-dimensional leaving us with two states corre- 1 sponding to helicity 2 . If we generalize these± considerations to higher spin Dirac type spinors, we find that the corresponding operator A has a zero kernel, ker A = 0, which can be easily verified in the momentum eigenspace p. In other words: the wave equation for massive D(j,0) D(0,j) spinor waveH functions leads for m 0 to ⊕ → a wave equation that has no solutions. This applies in particular to q-Maxwell spinors. Therefore, we need a different approach to find the q-Maxwell equations.

5.3 The q-Maxwell Equations 5.3.1 The q-Maxwell Equations in the Momentum Eigenspaces In this section we consider massless spinors ψj with the spinor index transforming according to a D(1,0) D(0,1) representation. According to the Clebsch-Gordan ⊕ series (2.6) this type of spinor is equivalent to considering an antisymmetric tensor F µν with two 4-vector indices. These are the types of spinor wave functions commonly used to describe the electromagnetic field, a massless field of spin-1. We start our calculations in the massless momentum eigenspace with Hp momentum eigenvalues p = (p0,p−,p+,p3)=(k, 0, 0,k) for some real parameter k. In Sec. 3.3.2 we have shown this eigenspace to be invariant under the little algebra 0, whose generators K, and NA have been defined in Eq. (3.100). Within theK little algebra acts only on the spinor index. The D(1,0) D(0,1) matrix Hp ⊕ representation of the generators are given by

1 ρ (J−) 0 −1 0 0 1 0 N− = q[2] , N+ = q [2] 1 , N3 = − 0 0 − 0 ρ (J+) 0 1 ρ1(K) 0 K = , 0 ρ1(K) (5.45)

1 where ρ is the vector representation map of q(su2). We seek a projector P = P 1 that projectsU onto an irreducible subrepresen- ⊗ tation of the little algebra. We write it in block form as A B P = , (5.46) CD 5.3 The q-Maxwell Equations 77

† where A, B, C, D are 3 3-matrices. We must have P0 = P0, so A and D must be Hermitian matrices and× C = B†. Recall from Eq. (3.105) that within an irreducible representation of we have N = 0. Therefore, we must have K0 ±

N±P = 0 (5.47) within the D(1,0) D(0,1) spinor representation. This leads to the conditions ⊕ 1 1 1 1 † ρ (J−) A =0 , ρ (J+) D =0 , ρ (J−) B =0 , ρ (J+) B =0 . (5.48)

To satisfy these conditions A, B, and D must be of the form

α 0 0 0 0 β 0 0 0 A = 0 0 0 , B = 0 0 0 ,D = 0 0 0 , (5.49)       0 0 0 0 0 0 0 0 δ       for α, δ real and β complex. Furthermore, P must project on an eigenvector of K. From this it follows that β = 0 and either α = 1, δ =0 or α = 0, δ = 1. To summarize, there are two possible projectors

1 0 0 P = , P = (5.50) L   R  0   0  1         projecting each on a irreducible one-dimensional representation of the little al- (1,0) gebra 0. The image of PL is part of the left handed D component while K (0,1) PR projects to the right handed D component of the spinor. Physically, this corresponds to left and right handed circular waves. We want to allow for parity transformations exchanging the left and right handed parts, so we need both parts

P = PL + PR . (5.51)

With the parity transformation included, the two dimensional space which P projects onto is irreducible.

5.3.2 Computing the q-Maxwell Equation We would like to find the q-Maxwell equation in the form of a first order differential equation

µ Aψ = CµP ψ =0 , (5.52) hoping that the operators Cµ can be chosen to act on the spinor index only, C = C 1. This wave equation has to fulﬁll condition (5.1): The q-Lorentz µ µ ⊗ 78 5. Free Wave Equations transform of a solution must again be a solution. For this, it would be suﬃcient but not necessary, if A were a scalar operator, as it has been the case for the q-Dirac equation and its zero mass limit, the q-Weyl equations. Recall from the last section, that as long as we do not include parity transformations, we must have two independent equations for the right and the left (1,0) handed part of the spinor, ψL carrying a D representation and ψR carrying a D(0,1) representation

ALψL =0 , ARψR =0 . (5.53) L µ R µ Let us try to choose AL = Cµ P and AR = Cµ P , so they commute with L R rotations. For this to be possible C0 , C0 must be scalars with respect to rotations L R while CA, CA must transform as 3-vectors. The only scalar operators within the D1-representation of rotations are multiples of the unit matrix, while every 3- 1 vector operator is proportional to ρ (JA). Hence, up to an overall constant factor our wave equations can be written as

0 1 A 0 1 A P + αL ρ (JA)P ψL =0 , P + αR ρ (JA)P ψR =0 , (5.54) whereαL, αR are constants. To determine these constants, we consider the wave equations in the momentum eigenspace, where they take the form

1 1 1+ αL ρ (J3) ψL =0 , 1+ αR ρ (J3) ψR =0 . (5.55) The space of solutions of each of these equations must equal the image of the projectors PL and PR, respectively. This requirement ﬁxes the constants to αL = −1 q and αR = q. Although this− determines our candidate for the q-Maxwell equation, condition (5.1) has yet to be checked for the boosts. Let ψ be an element of 0 ∈ Hp the momentum eigenspace, Pµψ0 = pµψ0, with pµ =(p0,p−,p+,p3)=(k, 0, 0,k). Using the commutation relations between boosts and momentum generators we ﬁnd

−1 −1 Pµ(aψ0)= q pµ(aψ0) , Pµ(bψ0)= q pµ(bψ0) (5.56a)

Pµ(cψ0)= qpµ(cψ0) , Pµ(dψ0)= qpµ(dψ0) . (5.56b) By induction it follows, that for any monomial in the boosts, h = aibjckdl, we k+l−i−j have Pµ(hψ0) = q pµ(hψ0). Thus, for any such ψ := hψ0, the wave equation (5.52) takes the form (C C )ψ =0 . (5.57) 0 − 3 Looking separately at the left and right handed part of ψ = ψL +ψR this equation writes out − − 00 0 ψL q[2] 0 0 ψR −2 3 2 3 0 q 0 ψL =0 , 0 q 0 ψR =0 , (5.58)  −1   +    + 0 0 q [2] ψL 0 00 ψR         5.3 The q-Maxwell Equations 79 which is equivalent to

3 + − 3 ψL = ψL =0 , ψR = ψR =0 . (5.59) If we now have a solution of Eq. (5.57), that is, a spinor ψ whose only non- − + vanishing components are ψL and ψR , could it happen that by boosting it gets other non-vanishing components, thus turning a solution into a non-solution? The answer to this question is no. We exemplify this, applying formula (5.5) for the action of the boost generator c on a left handed spinor,

′ A (1,0) A ′ A c ψL = ρ (c(1)) A c(2) ⊲ ψL ′ ′ (1,0) A ′ A (1,0) A ′ A = ρ (c) A a ⊲ ψL + ρ (d) A c ⊲ ψL 1 1 ′ 1 ′ 2 1 2 A ′ A 1 − 2 A ′ A = q λρ (FK ) A a ⊲ ψL + ρ (K ) A c ⊲ ψL − 0 1 0 a ⊲ ψ− q 0 0 c ⊲ ψ− 1 1 L L 2 2 3 3 = q λ[2] 0 0 1 a ⊲ ψL + 01 0 c ⊲ ψL −    +  −1  + 0 0 0 a ⊲ ψL 0 0 q c ⊲ ψL 1 1 − 2 2 3        q λ[2] a ⊲ ψL + qc⊲ψL − 1 1 + 2 2 3 = q λ[2] a ⊲ ψL + c ⊲ ψL , (5.60)  − −1 +  q c ⊲ ψL   3 + 3 + which clearly shows that, if ψL and ψL vanish, so do cψL and cψL . Similar calculations can be done for the other boost generators and right handed spinors. By induction we conclude, that if ψ0 p is a solution of Eq. (5.57) and i j k l ∈ H i j k l h = a b c d is a monomial in the boosts, h = a b c d , the spinor ψ = hψ0 will op be a solution, as well. The algebra of all boosts, SUq(2) , is generated as linear space by monomials, thus, hψ is a solution for any boost h SU (2)op. Since 0 ∈ q furthermore every q-Lorentz transformation can be written as a sum of products of rotations and boost, hψ0 is a solution for any q-Lorentz transformation h. We assume that the space of solutions, ker A, is an irreducible representation. This means in particular that the q-Lorentz algebra acts transitively on ker A, so any solution can be written as hψ0. Hence, the wave equations

ρ1(J )P Aψ = qP ψ , ρ1(J )P Aψ = q−1P ψ (5.61) A L − 0 L A R 0 R do indeed satisfy property (5.1). µ We want to write these equations, CµP ψ = 0 as q-diﬀerential equations µ C˜µ∂ ψ = 0, where C˜µ is deﬁned in Eq. (5.14). After lengthy calculations using Eqs. (A.42), (A.68), and (A.70) we get for the left and right handed part separately

′ C′ 1 ′ B ′ (1,0) Λ A 2 B ρ (JA ) C ρ (L+) A C = q εC A ′ − (5.62) 1 B (0,1) Λ A′ C −2 B ρ (JA′ ) C′ ρ (L ) A C = q εC A , + − 80 5. Free Wave Equations so the wave equations can be written as

∂~ ψ~ = iq−1∂ ψ~ , ∂~ ψ~ = iq ∂ ψ~ , (5.63) × L 0 L × R − 0 R ~ A ~ A where ψR =(ψR ), ψL =(ψL ), and where the cross product is deﬁned in Eq. (2.24). ~ A spinor ψL which is a solution to this equation must yet satisfy the mass zero condition. Using the identities (A.18) for the cross product, the commutation relations of the derivations ∂~ ∂~ = iλ∂ ∂~, and the wave equation (5.63), we × − 0 rewrite the mass zero condition as

0= ∂ ∂µψ~ =(∂2 ∂~ ∂~)ψ~ µ L 0 − · L = ∂2ψ~ (∂~ ∂~) ψ~ + ∂~ (∂~ ψ~ ) ∂~(∂~ ψ~ ) 0 L − × × L × × L − · L = ∂2ψ~ + iλ∂ (∂~ ψ~ )+ ∂~ (iq−1∂ ψ~ ) ∂~(∂~ ψ~ ) 0 L 0 × L × 0 L − · L = ∂2ψ~ q−1λ∂2ψ~ q−2∂2ψ~ ∂~(∂~ ψ~ ) 0 L − 0 L − 0 L − · L = ∂~(∂~ ψ~ ) . (5.64) − · L Contracting the wave equation with ∂~

∂~ (∂~ ψ~ )=(∂~ ∂~) ψ~ = iλ∂ (∂~ ψ~ ) = iq−1∂ (∂~ ψ~ ) , (5.65) · × L × · L − 0 · L 0 · L we see that ∂ (∂~ ψ~ ) = 0 if ψ~ is to satisfy the wave equation. Together with 0 · L L Eq. (5.64) this means that the mass zero condition is equivalent to ∂ (∂~ ψ~ ) = 0, µ · L that is, ∂~ ψ~L must be a constant number. In a momentum eigenspace we have ∂ (∂~ ψ~ )=· k(∂~ ψ~ ), so this constant number must be zero. The same reasoning 0 · L · L applies for the right handed spinor ψ~R. We conclude that the wave equations (5.63) together with the mass zero con- µ dition ∂µ∂ ψ = 0 are equivalent to

∂~ ψ~ = iq−1∂ ψ~ , ∂~ ψ~ = 0 (5.66) × L 0 L · L ∂~ ψ~ = iq ∂ ψ~ , ∂~ ψ~ =0 , (5.67) × R − 0 R · R which we will call the q-Maxwell equations.

5.3.3 The q-Electromagnetic Field Finally, we write the q-Maxwell equations in a more familiar form, that is, in terms of the q-deformed electric and magnetic fields. In the undeformed case the electric and magnetic fields can — up to constant factors — be characterized within the D(1,0) D(0,1) representation as eigenstates of the parity operator. ⊕ The electric field should transform like a polar vector E~ = E~ , while the magnetic field must be an axial vector B~ = B~ . Recall, thatP the parity− operator P 5.3 The q-Maxwell Equations 81

acts on q-spinors by exchanging the left and the right handed parts ψL = ψR, Pψ = ψ . This fixes the fields P P R L E~ = i(ψ~ ψ~ ) , B~ = ψ~ + ψ~ (5.68) R − L R L up to constant factors which have been chosen to give the right undeformed limit. Spinor conjugation of the fields is now the same as ordinary conjugation E¯A = (EA)∗, B¯A =(BA)∗. In terms of these fields, the q-Maxwell equations (5.66) take the form ∂~ E~ = 1 [2] ∂ B~ 1 iλ ∂ E,~ ∂~ E~ = 0 (5.69) × 2 0 − 2 0 · ∂~ B~ = 1 [2] ∂ E~ 1 iλ ∂ B,~ ∂~ B~ =0 . (5.70) × − 2 0 − 2 0 · We would also like to express the q-Maxwell equations in terms of a field strength tensor F µν. According to the Clebsch-Gordan series the left and right chiral 3- vectors ψL and ψR can be each identified with a 4-vector matrix. If we replace in ψ = e ψC the spinor basis e with E1,0 from formula (A.26), L C ⊗ L C C,0 ψ = e ψC =(E E εAB + qE E q−1E E ) ψC L C ⊗ L A ⊗ B C 0 ⊗ C − C ⊗ 0 ⊗ L =(E E ) F µν , (5.71) µ ⊗ ν ⊗ L this defines the matrix F 00 F 0N 0 qψN F µν := L L = L , (5.72) L F M0 F MN q−1ψM εMN ψC L L − L C L where M, N run through , +, 3 . In the same manner we obtain for the right {− } handed part 0 q−1ψN F µν := − R . (5.73) R qψM εMN ψC R C R µν In terms of these matrices the q-Maxwell equations (5.66) take the form ∂ν FL =0 µν and ∂νFR = 0. This suggests to introduce the field strength tensor and its dual F µν := i(F µν + F µν) , F˜µν := i(F µν F µν) , (5.74) L R L − R where the factor i is needed for the right undeformed limit. In terms of the electric and the magnetic field, we have 1 N 1 N µν 0 2 [2]E + 2 iλB F := 1 M 1 M − MN C [2]E + iλB iε C B 2 2 (5.75) 0 1 [2]iBN 1 λEN F˜µν := 2 2 . 1 [2]iBM 1 λEM εMN− EC − 2 − 2 − C The q-Maxwell equations become

µν µν ∂νF =0 , ∂νF˜ =0 , (5.76) in complete analogy to the undeformed case. Appendix A Useful Formulas

A.1 Clebsch-Gordan Coeﬃcients

A.1.1 Clebsch-Gordan and Racah Coefficients for Uq(su2) We first list some formulas which enable us to calculate some q-Clebsch-Gordan coefficients explicitly [62, 63]:

′ ′ Cq(0, j, j 0, m, m )= δmm′ δjj′ | 1 1 1 1 1 ±(j∓m)/2 1 −1 2 Cq(j, 2 , j + 2 m, 2 , m 2 )= q ([j m + 2 ][2j + 1] ) (A.1) | ± ± ± 1 C (j, 1 , j 1 m, 1 , m 1 )= q∓(j±m+1)/2([j m][2j + 1]−1) 2 q 2 − 2 | ± 2 ± 2 ∓ ∓ For C (1, j, j + ∆j ∆m, m, m + ∆m) we have the formulas q |

∆j ∆m = 1 ∆m = 0 ∆m = +1 − 1 qm−j−1 [j+m][j+m−1] qm [2][j+m][j−m] qm+j+1 [j−m][j−m−1] − [2j+1][2j] − [2j+1][2j] [2j+1][2j] q (j+1)q −(j+1) q 0 qm−1 [2][j+m][j−m+1] qm q [j−m]−q [j+m] qm+1 [2][j+m+1][j−m] − [2j+2][2j] √[2j+2][2j] [2j+2][2j] q q m+j [j−m+2][j−m+1] m [2][j+m+1][j−m+1] m−j [j+m+2][j+m+1] +1 q [2j+2][2j+1] q [2j+2][2j+1] q [2j+2][2j+1] q q q The q-Clebsch-Gordan coeﬃcients obey the symmety

′ ′ ′ j′−j ν [2j + 1] ′ ′ Cq(n, j, j ν, m, m )=( 1) ( q) Cq(n, j , j ν, m , m) . (A.2) | − − s [2j + 1] | − For the q-Racah coeﬃcients we have

′ ′ ′ j′+j [2j + 1] [3] Rq(1, 1, j 0, j , j )=( 1) . (A.3) − | − s [2j + 1] p For [4] R (1, 1, j 1, j′, j′′) there are the formulas − [2] q | q A.1 Clebsch-Gordan Coeﬃcients 83

j′′ = j 1 j′′ = j j′′ = j +1 − j′ = j 1 [2j−2] [2j+2][2j−1] 0 − [2j] [2j+1][2j] q q j′ = j [2j+2] [2j]−[2j+2] [2j] − [2j] √[2j+2][2j] [2j+2] q q j′ = j +1 0 [2j+3][2j] [2j+4] − [2j+2][2j+1] − [2j+2] q q A.1.2 Metric and Epsilon Tensor We deﬁne the 3-metric and the epsilon tensor as1

AB AB [4] g := [3]Cq(1, 1, 0 A, B, 0) , ε C = Cq(1, 1, 1 A,B,C) , (A.4) − | −s[2] | p where the capital roman indices run through 1, 0, 1 = , 3, + . The positions of the indices are chosen such that the basis{− vectors} are{− written} with lower indices. From this deﬁnition it is clear that the projectors on the subspaces on the right hand side of the Clebsch-Gordan series

D1 D1 = D0 D1 D3, (A.5) ⊗ ∼ ⊕ ⊕ that we denote by P0, P1, P3, can be written as

AB −1 AB P0 CD = [3] g gCD AB −1 ABX P1 CD = [2][4] ε εDCX (A.6) PAB = δAδB PAB PAB , 3 CD C D − 0 CD − 1 CD AB where the projectors act on lower indices, P⊲EC ED := EAEBP CD. The nonzero values of the metric are

g−+ = q−1 , g+− = q, g33 =1 . (A.7) − − AB By deﬁnition gAB is the inverse of g

BC C CB gABg = δA = g gBA , (A.8)

AB implying gAB = g . This means that we can not raise and lower the indices of the metric as usual. Instead, we get

A′B′ gAA′ gBB′ g = gBA . (A.9)

1 AB AB C Metric and epsilon tensor, gAB and ε C , as deﬁned here correspond to g and qεBA in [43]. 84 A. Useful Formulas

The nonzero values of the epsilon tensor are

ε−3 = q−1 ε3− = q (A.10a) − − − ε−+ =1 ε+− = 1 ε33 = λ (A.10b) 3 3 − 3 − ε3+ = q−1 ε+3 = q . (A.10c) + + − B A′B Lowering the ﬁrst index as usual by εA C := gAA′ ε C we get

3 − ε+ − = 1 ε3 − = q (A.11a) + − − −−1 3 ε+ 3 = q ε− 3 = q ε3 3 = λ (A.11b) + −−1 3 − ε3 + = q ε− + =1 . (A.11c)

Lowering the second index

− −1 3 2 ε 3− = q ε +− = q (A.12a) ε− = q−1 ε+ = q ε3 = λ (A.12b) −3 − +3 33 − ε3 = q−2 ε+ = q . (A.12c) −+ − 3+ − A′ With all indices down εABC := gAA′ ε BC

ε = 1 ε = q2 (A.13a) +3− − 3+− ε =1 ε = 1 ε = λ (A.13b) +−3 −+3 − 333 − ε = q−2 ε =1 . (A.13c) 3−+ − −3+ Various contractions of ε-tensor and metric yield useful identities

AB′C CA AB C′C BCA ε gB′B = ε B , ε C′ g = ε A′B′C C A′A C′C CA ε gA′AgB′B = εB A , εA′BC′ g g = ε B g εABC =0 , g εABC =0 , ε gBA =0 , ε gAC =0 AB CA ABC ABC (A.14) AXB BA X BAX ε εCXD = ε X εC D = ε εDCX B AC −1 B BAD ADB −1 D εA C ε D = [4][2] δD , εABC ε = εBCA ε = [4][2] δC AB XC AB C AX BC A BC ε X ε D + g δD = ε Dε X + δDg .

There are relations between ε-tensors with the same index in an upper and a lower position

ACB B AC A CB εABC = ε , εA C = ε B , ε BC = εA . (A.15)

With the metric and the epsilon tensor we can define a scalar and a vector ∗ product. Note, that if we defined real coordinates by X1 := i(X+ X+), X2 := X + X∗ we would get, e.g., ε123 = qi. In the limit q 1 our− epsilon tensor + + − → A.1 Clebsch-Gordan Coefficients 85 will tend to i times the usual undeformed epsilon tensor. We therefore define − for 3-vector operators XA and YB X~ Y~ := gABX Y , (X~ Y~ ) := i X Y εAB , (A.16) · A B × C A B C where we use arrows to indicate the 3-vector operators. Raising and lowering the indices we get X~ Y~ := g XAY B , (X~ Y~ )C := i XAY Bε C . (A.17) · BA × B A With this notation some of the identities (A.14) take on a very intuitive form X~ (Y~ Z~)=(X~ Y~ ) Z~ · × × · (A.18) (X~ Y~ ) Z~ (X~ Y~ )Z~ = X~ (Y~ Z~) X~ (Y~ Z~) , × × − · × × − · from which more relations can be deduced very easily. Finally, we apply Eq. (2.21) to the scalar and the vector product

′ [2j′ + 1] j X~ Y~ j = ( 1)j +j j X~ j′ j′ Y~ j (A.19a) h k · k i ′ − s [2j + 1] h k k ih k k i Xj [2j 2] j 1 X~ Y~ j = i − j 1 X~ j 1 j 1 Y~ j h − k × k i s [2j] h − k k − ih − k k i [2j + 2] i j 1 X~ j j Y~ j (A.19b) − s [2j] h − k k ih k k i

[2j + 2][2j 1] j X~ Y~ j = i − j X~ j 1 j 1 Y~ j h k × k i s [2j + 1][2j] h k k − ih − k k i [2j] [2j + 2] +i − j X~ j j Y~ j [2j + 2][2j] h k k ih k k i p[2j + 3][2j] i j X~ j +1 j +1 Y~ j (A.19c) − s[2j + 2][2j + 1] h k k ih k k i

[2j] j +1 X~ Y~ j = i j +1 X~ j j Y~ j h k × k i s[2j + 2] h k k ih k k i [2j + 4] i j +1 X~ j +1 j Y~ j . (A.19d) − s[2j + 2] h k k ih k k i If furthermore there is a -structure X∗ = Y A, this implies for the reduced matrix ∗ A elements of a -representation ∗ ′ [2j + 1] j′ X~ j =( 1)j −j j Y~ j′ . (A.19e) h k k i − s[2j′ + 1] h k k i 86 A. Useful Formulas

A.1.3 Clebsch-Gordan Coefficients for the q-Lorentz Algebra The Clebsch-Gordan Coefficients for the q-Lorentz algebra can be read off the formula for the basis vectors of the irreducible subrepresentations

(k ,k ), (n , n ) = C (j , j′ ,k m , b, n )C (j , j′ ,k a, m′ , n ) | 1 2 1 2 i q 1 1 1 | 1 1 q 2 2 2 | 2 2 m m′ (XR−1) 2 1 (j , j ), (m , m ) (j′ , j′ ), (m′ , m′ ) , (A.20) × ab | 1 2 1 2 i⊗| 1 2 1 2 i As the R-matrix is in general not unitary, these basis vectors have yet to be normalized. We are in particular interested in the q-Clebsch-Gordan coeﬃcients for the decomposition of a tensor product of two vector representations

( 1 , 1 ) ( 1 , 1 ) (0,0) (1,0) (0,1) (1,1) D 2 2 D 2 2 = D D D D . (A.21) ⊗ ∼ ⊕ ⊕ ⊕ For a more compact notation we write

Ej1j2 := (j , j ), (m , m ) , E := ( 1 , 1 ), (a, b) , (A.22) m1m2 | 1 2 1 2 i ab | 2 2 i 1 1 where a, b run through 2 , 2 = , + . We get for the unnormalized basis vectors of the D(1,0) subrepresentation{− } {− } E1,0 = qE E q−1E E −1,0 −+ ⊗ −− − −− ⊗ −+ 1,0 E = E++ E−− E−− E++ + λE−+ E−+ 0,0 ⊗ − ⊗ ⊗ (A.23) + E E q−2E E −+ ⊗ +− − +− ⊗ −+ E1,0 = E E E E + λE E , +1,0 ++ ⊗ +− − +− ⊗ ++ ++ ⊗ −+ for the D(0,1) subrepresentation E0,1 = E E E E + λE E 0,−1 +− ⊗ −+ − −− ⊗ +− −+ ⊗ −− 0,1 E = E++ E−− E−− E++ + λE−+ E−+ 0,0 ⊗ − ⊗ ⊗ (A.24) + E E q−2E E +− ⊗ −+ − −+ ⊗ +− E0,1 = qE E q−1E E , 0,+1 ++ ⊗ −+ − −+ ⊗ ++ and for the D(0,0) subrepresentation

0,0 −1 −1 E = qE++ E−− + q E−− E++ q E−+ E+− 0,0 ⊗ ⊗ − ⊗ (A.25) q−1E E q−1λE E . − +− ⊗ +− − −+ ⊗ −+ Expressed in terms of a 4-vector basis we find bases for the D(1,0), D(0,1), and D(0,0) subrepresentations E1,0 = E E εAB + qE E q−1E E C,0 A ⊗ B C 0 ⊗ C − C ⊗ 0 E0,1 = E E εAB + qE E q−1E E (A.26) 0,C A ⊗ B C C ⊗ 0 − 0 ⊗ C E0,0 = E E ηµν , 0,0 µ ⊗ ν A.1 Clebsch-Gordan Coefficients 87 which are neither orthogonal nor normalized. The last equation defines up to a constant factor the 4-vector metric ηµν whose non-zero values are

η00 =1 , η−+ = q−1 , η+− = q , η33 = 1 , (A.27) − which means in particular that ηAB = gAB. Let us denote the projectors on the ( 1 , 1 ) − 2 subrepresentations of the D 2 2 representation in an obvious notation by

1= P(0,0) + P(1,0) + P(0,1) + P(1,1) . (A.28)

The projectors on the symmetric and antisymmetric part are denoted by

PS := P(0,0) + P(1,1) , PA := P(1,0) + P(0,1) . (A.29)

These projectors can be determined from the bases of the corresponding spaces, which we just computed. Note however that D(1,0) and D(0,1) are not mutually orthogonal, so we have to project on D(1,0) along D(0,1) and vice versa. We obtain for the trace part

ab −2 ab (P(0,0)) cd = [2] η ηcd , (A.30) for the left chiral and right chiral part:

2 ab 2 ab [2] (P(1,0)) cd = [2] (P(0,1)) cd = C0 0DCD C0 0DCD A0 δA q−2δA q−1ε A A0 δA q2δA qε A C − D − C D C − D C D 0B q2δB δB qε B 0B q−2δB δB q−1ε B − C D C D − C D − C D AB qεAB q−1εAB εAB ε X AB q−1εAB qεAB εAB ε X − C D X C D C − D X C D For the anti-symmetrizer this yields

2 ab [2] (PA) cd = C0 0DCD A0 2δA [4][2]−1δA λε A C − D C D 0B [4][2]−1δB 2δB λε B − C D C D AB λεAB λεAB 2εAB ε X − C − D X C D The traceless symmetric part is given by Eq. (A.28).

2 Elsewhere [43] the same projectors have been denoted by PT , P+, P−, PS , in that order. 88 A. Useful Formulas

A.2 Representations

A.2.1 Representations of Uq(su2) The action of the generators within the Dj representation of (su ) is given by Uq 2 E j, m = q(m+1) [j + m + 1][j m] j, m +1 | i − | i F j, m = q−m p[j + m][j m + 1] j, m 1 (A.31) | i − | − i K j, m = q2m j, m . | i |p i For for the vectorial generators this means

J j, m = [2]−1 [2j + 2][2j] C (1, j, j A, m, m + A) j, m + A . (A.32) A| i − q | | i The value of the Casimirp W within such a representation is given by

ρj(W ) = [2]−1 q(2j+1) + q−(2j+1) . (A.33)

1 For j = 2 the generators are represented by

− 1 −1 0 0 0 q 2 q 0 E := 1 , F := , K := , (A.34) q 2 0 0 0 0 q with respect to the , + basis. The representation of the vector generators J {− } A is proportional to the q-Pauli matrices

1 1 σ = [2] ρ 2 (J ) , σ˜ = [2] ρ 2 (SJ ) , (A.35) A A A − A where

− 1 1 0 q 2 1 0 0 q 0 2 2 σ− = [2] , σ+ = [2] 1 , σ3 = − −1 (A.36) 0 0 q 2 0 0 q − 1 −1 1 0 q 2 1 0 0 q 0 2 2 σ˜− = [2] , σ˜+ = [2] − 1 , σ˜3 = − , (A.37) 0 0 q 2 0 0 q − 1 1 with respect to the 2 , 2 = , + basis. The q-Pauli matrices satisfy the relations {− } {− }

AB −1 C σA σB ε C = [4][2] σC , σAσB = gAB + σC εA B (A.38) σ˜ σ˜ εBA = [4][2]−1 σ˜ , σ˜ σ˜ = g σ˜ ε C , (A.39) A B C − C A B BA − C B A For the j = 1 vector representations we get

0 0 0 01 0 q−2 0 0 1 1 −1 E := [2] 2 1 0 0 , F := [2] 2 0 0 q , K := 0 10 (A.40)       0 q 0 00 0 0 0 q2       A.2 Representations 89

0 q−1 0 0 0 0 q 0 0 − J− := 0 0 1 , J+ := 1 0 0 , J3 := 0 λ 0 (A.41) 0 0 0 −0 q 0  0− 0 q−1 − with respect to the 1, 0, 1 = , 3, + basis. The matrix representations of the vector generator{− is proportional} {− to the} epsilon tensor, 1 B B ρ (JA) C = εA C . (A.42)

A.2.2 Representations of the q-Lorentz Algebra

The representation maps for the D(j1,j2) representations of the q-Lorentz algebra, (sl (C)), are composed of the representation maps of (su ) according to Uq 2 Uq 2 ρ(j1,j2) := ρj1 ρj2 . Particularly simple are the chiral representations D(j,0) and (0,j) ⊗ D . For any rotations l q(su2) and for the boosts as deﬁned in Eq. (A.63) we get ∈ U ρ(j,0)(l)= ρj(l)= ρ(0,j)(l) 1 1 (A.43) (j,0) a j 2 a (0,j) a j 2 a ρ (B b)= ρ (L−) b , ρ (B b)= ρ (L+) b . ( 1 , 1 ) If we denote the basis of a D 2 2 representation as in Eq. (3.1) by E E A B E = −− −+ =: , (A.44) ab E E CD +− ++ we get for the action A B CD A B B 0 E 1 ⊲ = , 1 E⊲ = ⊗ CD 0 0 ⊗ CD D 0 A B 0 0 A B 0 A F 1 ⊲ = , 1 F ⊲ = (A.45) ⊗ CD A B ⊗ CD 0 C A B q−1A q−1B A B q−1A qB K 1 ⊲ = , 1 K⊲ = . ⊗ CD qC qD ⊗ CD q−1C qD For the boost generators (A.63) this means in particular a b A q−1λB a b q−1B 0 ⊲ A = , ⊲ B = (A.46a) c d 0 A c d 0 qB a b qC λD a b D 0 ⊲C = , ⊲ D = (A.46b) c d qλA q−1C λ2B c d λB D − − − ( 1 , 1 ) In terms of the 4-vector basis Eµ of D 2 2 , deﬁned as in Eq. (3.10) by −1 −1 1 − 1 E = q [2] (q 2 C q 2 B) 0 − − 1 E− = [2] 2 A (A.47) − 1 E+ = [2] 2 D −1 − 1 1 E3 = [2] (q 2 C + q 2 B) . 90 A. Useful Formulas the action becomes

[4] 1 1 −1 −1 − 2 − 2 a b [2] [2] E0 + q λE3 q λ[2] E+ ⊲ E0 = 1 − 1 −1 [4] c d  q 2 λ[2] 2 E [2] E qλE  − − [2] 0 − 3  − 1 − 1  a b E− q 2 λ[2] 2 (E3 E0) ⊲ E− = − c d 0 E − (A.48) a b E+ 0 ⊲ E+ = 1 − 1 c d q 2 λ[2] 2 (E E ) E − 3 − 0 + −1 − 1 − 1 a b [2] (2E3 + qλE0) q 2 λ[2] 2 E+ ⊲ E3 = 1 − 1 −1 −1 c d q 2 λ[2] 2 E [2] (2E q λE ) − − 3 − 0 Now we can calculate the 4-vector matrix representation Λ deﬁned by

µ′ h ⊲ Eµ = Eµ′ Λ(h) µ (A.49) for all q-Lorentz transformations h. For the rotations l q(su2) we get by construction of the 4-vector basis ∈ U ρ0(l) 0 Λ(l)= . (A.50) 0 ρ1(l) For the boost we calculate [4][2]−2 0 0 qλ[2]−1 0 1 0 0 − 0 10 0 − 1 − 1 0 0 00 Λ(a)=   , Λ(b)= q 2 λ[2] 2   0 01 0 1 0 01 q−1λ[2]−1 0 0 2[2]−1  0 1 00      0 0 1 0  [4][2]−2 0 0 q−1λ[2]−1 − − 1 − 1 10 0 1 0 10 0 Λ(c)= q 2 λ[2] 2   , Λ(d)=   , − 00 0 0 0 01 0 00 1 0  qλ[2]−1 0 0 2[2]−1    −     (A.51) with respect to the 0, , +, 3 basis. { − } A.3 R-matrices For a Hopf algebra H a universal -matrix is an invertible element H H, R R ∈ ⊗ which we will also write in a Sweedler like notation as := , with R R[1] ⊗ R[2] (τ ∆)(h)= ∆(h) −1 ◦ R R (A.52) (∆ id)( )= , (id ∆)( )= , ⊗ R R13R23 ⊗ R R13R12 A.3 -matrices 91 R where the indices indicate the position of the tensor factors, 13 := [1] 1 [2] etc. If there is a -structure on H the -matrix is said to beR real ifR ∗⊗∗⊗ =⊗R ∗ R R R21 and anti-real if ∗⊗∗ = −1. There are some useful properties of that can be deduced from Eqs.R (A.52):R R

= , (ε id)( ) = 1 (id ε)( )=1 R12R13R23 R23R13R12 ⊗ R ⊗ R (A.53) (S id)( )= −1 , (id S)( −1)= , (S S)( )= . ⊗ R R ⊗ R R ⊗ R R

A.3.1 The R-Matrix of Uq(su2) There is a universal -matrix for (su ), R Uq 2 ∞ = q(H⊗H)/2 R (q)(En F n) (A.54) R n ⊗ n=0 X which is not an element q(su2) q(su2) proper, since it is described as an inﬁnite power series. For ourU purposes⊗ U this does not raise serious problems. This ′ -matrix is real. For representations ρj, ρj of (su ) we can deﬁne R-matrices R Uq 2 and a variant, the Rˆ-matrices, by

′ ′ ′ ′ R(j,j ) := (ρj ρj )( ) , (Rˆ(j,j ))ab := (R(j,j ))ba . (A.55) ⊗ R cd cd Traditionally, the R-matrices are normalized diﬀerently. We will use

1 1 1 2 ( 2 , 2 ) −2 (1,1) Rsu2 := q R , Rso3 := q R . (A.56) Explicitly, we get q 0 0 0 ab 0 100 (Rsu ) cd =   , (A.57) 2 0 λ 1 0 0 0 0 q     with respect to the basis , +, + , ++ , and {−− − − } AB B A −3 BA −2 BAX (Rso ) CD = δ δ q λg gCD q ε εDCX 3 C D − − (A.58) (R−1 )AB = δAδB q3λgABg q2εABX ε . so3 CD D C − DC − CDX This means that we have a projector decomposition

Rˆ =1 q−3λ[3]−1P q−2[4][2]−1P = q−6P q−4P + P . (A.59) so3 − 0 − 1 − 0 − 1 3 Applying a representation to one half of the -matrix only leads to the deﬁnition of the L-matrices R

(Lj )a := ρj( )a , (Lj )a := ρj( −1)a −1 . (A.60) + b R[1] R[2] b − b R[1] b R[2] 92 A. Useful Formulas

1 We calculate the L-matrices for j = 2 and j = 1, explicitly. 1 1 1 1 1 − 2 − 2 − 2 1 2 2 K q λK E 2 K 0 L+ = 1 , L− = 1 1 − 1 (A.61) 0 K 2 q 2 λFK 2 K 2 − −1 1 −1 2 −1 2 K λ[2] 2 K E λ K E K 0 0 1 1 −1 1 1 L+ = 0 1 q λ[2] 2 E , L− = λ[2] 2 FK 1 0   − 2 2 1 −1 0 0 K λ F K qλ[2] 2 F K −    (A.62) These results are being used in Eq. (2.50) to calculate the boost generators deﬁned as 1 a 1 b a b Ba := L 2 L 2 =: , (A.63) c − b ⊗ + c c d which yields 1 − 1 − 1 1 − 1 a b K 2 K 2 q 2 λK 2 K 2 E = 1 ⊗ 1 − 1 − 1 1 ⊗2 1 − 1 . (A.64) c d q 2 λFK 2 K 2 K 2 K 2 λ FK 2 K 2 E − ⊗ ⊗ − ⊗ A.3.2 The R-Matrices of the q-Lorentz Algebra There are two universal -matrices of the q-Lorentz algebra, which are composed of the -matrix of (slR) according to R Uq 2 = −1 −1 , = −1 . (A.65) RI R41 R31 R24R23 RII R41 R13R24R23 is anti-real while is real. Their vector representations are normalized as RI RII R := (Λ Λ)( ) , R := q(Λ Λ)( ) , (A.66) I ⊗ RI II ⊗ RII where Λ is the 4-vector representation map of the q-Lorentz algebra. These matrices can be decomposed into projectors −2 2 RˆI = P(0,0) q P(1,0) q P(0,1) + P(1,1) − − (A.67) Rˆ = q−2P P P + q2P . II (0,0) − (1,0) − (0,1) (1,1) The L -matrix of has a simple form: + RI Λ a a 1 0 LI+ b := I[1] Λ( I[2]) b = A , (A.68) R R 0 t B A op where t B is the vector corepresentation matrix of SUq(2) , 2 1 1 2 a q 2 [2] 2 ab b 1 1 1 1 t = q 2 [2] 2 ac (1 + [2]bc) q 2 [2] 2 bd (A.69)  2 1 1 2  c q 2 [2] 2 cd d with respect to the basis 1, 0, 1 = , 3, + . For chiral representations we {− } {− } get (j,0) A j 1 A (0,j) A j 1 A ρ (t B)= ρ (L−) B , ρ (t B)= ρ (L+) B . (A.70) Bibliography

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